The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
The document discusses graphing sine curve functions. It explains that a basic sine curve has the equation y = sin(x) with a period of 2π units. More generally, a sine curve has the equation y = a sin(bx + c) where the period is 2π/b units, the amplitude is a units, and c shifts the curve left or right. It provides an example of y = 5 sin(9x - π/2) with a period of 2π/9 units, amplitude of 5 units, and a right shift of π/18.
The document discusses trigonometric functions, arcs, sectors, and formulas related to them. It defines an arc as a segment of a circle and provides the formula for calculating the length of an arc as l = rθ. It also defines a sector as a region bounded by two radii and an arc and gives the formula for calculating the area of a sector as A = 1/2 r^2θ. An example problem demonstrates using these formulas to calculate the length of an arc and area of a sector for a given diagram.
The document discusses the geometric representation of complex numbers using vectors on the Argand diagram. It explains that complex numbers can be represented as vectors, with the real part of the number as the x-coordinate and imaginary part as the y-coordinate of the vector. Addition and subtraction of complex numbers corresponds to placing the vectors head to tail and head to head, respectively. Properties like the parallelogram law and triangle inequality are demonstrated. Multiplication of complex numbers is shown to be equivalent to multiplying the magnitudes and adding the arguments of the vectors.
The document defines and provides examples of absolute value. Absolute value is the distance of a number from 0, regardless of its positive or negative sign. It only considers magnitude, not direction. Examples are provided to illustrate calculating the absolute value of numbers and solving absolute value equations and inequalities, which involve treating the absolute value expression as both positive and negative terms.
The document discusses solving inequalities involving quadratic and rational expressions. For quadratic inequalities, it explains how to factorize, set each factor equal to zero to find critical values, and use these to determine intervals where the parabola is above or below the x-axis. For rational inequalities, it outlines steps to find where the denominator is zero, solve the resulting equality, plot critical values on a number line, and test intervals to determine the solution set. The document provides examples demonstrating these techniques.
The document discusses finding the greatest term in polynomial expansions. It provides an example of finding the greatest coefficient in the expansion of (2 + 3x)^20. Through algebraic steps, it is shown that the greatest coefficient is T13 = 20C12 * 2831^2. It then gives an example of finding the greatest term in the expansion of (3x - 4)^15 when x = 1/2. After setting up expressions for the terms Tk+1 and Tk, it derives an inequality to determine the value of k that makes Tk+1 the greatest term.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
The document discusses graphing sine curve functions. It explains that a basic sine curve has the equation y = sin(x) with a period of 2π units. More generally, a sine curve has the equation y = a sin(bx + c) where the period is 2π/b units, the amplitude is a units, and c shifts the curve left or right. It provides an example of y = 5 sin(9x - π/2) with a period of 2π/9 units, amplitude of 5 units, and a right shift of π/18.
The document discusses trigonometric functions, arcs, sectors, and formulas related to them. It defines an arc as a segment of a circle and provides the formula for calculating the length of an arc as l = rθ. It also defines a sector as a region bounded by two radii and an arc and gives the formula for calculating the area of a sector as A = 1/2 r^2θ. An example problem demonstrates using these formulas to calculate the length of an arc and area of a sector for a given diagram.
The document discusses the geometric representation of complex numbers using vectors on the Argand diagram. It explains that complex numbers can be represented as vectors, with the real part of the number as the x-coordinate and imaginary part as the y-coordinate of the vector. Addition and subtraction of complex numbers corresponds to placing the vectors head to tail and head to head, respectively. Properties like the parallelogram law and triangle inequality are demonstrated. Multiplication of complex numbers is shown to be equivalent to multiplying the magnitudes and adding the arguments of the vectors.
The document defines and provides examples of absolute value. Absolute value is the distance of a number from 0, regardless of its positive or negative sign. It only considers magnitude, not direction. Examples are provided to illustrate calculating the absolute value of numbers and solving absolute value equations and inequalities, which involve treating the absolute value expression as both positive and negative terms.
The document discusses solving inequalities involving quadratic and rational expressions. For quadratic inequalities, it explains how to factorize, set each factor equal to zero to find critical values, and use these to determine intervals where the parabola is above or below the x-axis. For rational inequalities, it outlines steps to find where the denominator is zero, solve the resulting equality, plot critical values on a number line, and test intervals to determine the solution set. The document provides examples demonstrating these techniques.
The document discusses finding the greatest term in polynomial expansions. It provides an example of finding the greatest coefficient in the expansion of (2 + 3x)^20. Through algebraic steps, it is shown that the greatest coefficient is T13 = 20C12 * 2831^2. It then gives an example of finding the greatest term in the expansion of (3x - 4)^15 when x = 1/2. After setting up expressions for the terms Tk+1 and Tk, it derives an inequality to determine the value of k that makes Tk+1 the greatest term.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
6. Differentiating Trig
lim sin x 0 Area AOC Area Sector OAC
x 0
lim cos x 1
x 0
lim tan x 0
x 0
C
x A
O 1
7. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
lim cos x 1
x 0
lim tan x 0 B
x 0
C
tan x
x A
O 1
8. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B
x 0
C
tan x
x A
O 1
9. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B sin x x tan x
x 0
C
tan x
x A
O 1
10. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B sin x x tan x
x 0
sin x x tan x
sin x sin x sin x
C
tan x
x A
O 1
11. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B sin x x tan x
x 0
sin x x tan x
sin x sin x sin x
C
tan x
x 1
1
sin x cos x
x A
O 1
12. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B sin x x tan x
x 0
sin x x tan x
sin x sin x sin x
C
tan x
x 1
1
sin x cos x
x A as x 0
O 1 x
1 1
sin x
13. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B sin x x tan x
x 0
sin x x tan x
sin x sin x sin x
C
tan x
x 1
1
sin x cos x
x A as x 0
O 1 x
1 1
sin x
x
lim 1
x 0 sin x
16. 5x
e.g. i lim 1 ii lim
x
x 0 sin 5 x x 0 sin 3 x
17. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
18. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
1
3
19. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
3
20. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
21. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
sin x cosh cos x sinh sin x
lim
x 0 h
22. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
sin x cosh cos x sinh sin x
lim
x 0 h
cosh 1
lim cos x
sinh
sin x
x 0 h h
23. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
sin x cosh cos x sinh sin x
lim
x 0 h
cosh 1
lim cos x
sinh
sin x
x 0 h h
h
2 cos 2 2
sinh 2
lim cos x sin x
x 0 h
h
24. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
sin x cosh cos x sinh sin x
lim
x 0 h
cosh 1
lim cos x
sinh
sin x
x 0 h h
h
2 cos 2 2
sinh 2 cos 2 2 cos 2 1
lim cos x sin x
x 0 h
h
25. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
sin x cosh cos x sinh sin x
lim
x 0 h
cosh 1
lim cos x
sinh
sin x
x 0 h h
h
2 cos 2 2
sinh 2 cos 2 2 cos 2 1
lim cos x sin x
x 0 h
h
2 h
sinh 2 sin
lim cos x sin x 2
x 0 h h
26. 2 h
sin
lim cos x
sinh
sin x h 2
x 0 h
2
27. 2 h
sin
lim cos x
sinh
sin x h 2
x 0 h
2
h
sinh sin
h
lim cos x sin x h 2 sin
x 0 h
2
2
28. 2 h
sin
lim cos x
sinh
sin x h 2
x 0 h
2
h
sinh sin
h
lim cos x sin x h 2 sin
x 0 h
2
2
cos x 1 sin x 0
29. 2 h
sin
lim cos x
sinh
sin x h 2
x 0 h
2
h
sinh sin
h
lim cos x sin x h 2 sin
x 0 h
2
2
cos x 1 sin x 0
cos x
30. 2 h
sinh sin
2
lim cos x sin x h
x 0 h
2
h
sinh sin
h
2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0
cos x
31. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
32. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
33. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
y sin x
2
34. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
y sin x
2
cos x
dy
dx 2
35. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
y sin x
2
cos x
dy
dx 2
sin x
36. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
y sin x
2 y cos f x
cos x
dy
dx 2
sin x
37. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
y sin x
2 y cos f x
cos x
dy
f x sin f x
dy
dx 2 dx
sin x
40. y tan x
sin x
y
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
41. y tan x
sin x
y
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
42. y tan x
sin x
y
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
43. y tan x
sin x
y
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
44. y tan x e.g. i y sin x 3
sin x
y
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
45. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
46. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
47. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos 2 x sin 2 x 2 sec 2
dx x x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
48. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
49. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x
cos 2 x dy sin x
sec x
2
dx cos x
y tan f x
dy
f x sec 2 f x
dx
50. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x
cos 2 x dy sin x
sec x
2
dx cos x
tan x
y tan f x
dy
f x sec 2 f x
dx
51. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x iv y tan 5 x
cos 2 x dy sin x
sec x
2
dx cos x
tan x
y tan f x
dy
f x sec 2 f x
dx
52. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x iv y tan 5 x
cos 2 x dy sin x dy
5 tan 4 x sec 2 x
sec 2 x dx cos x dx
tan x
y tan f x
dy
f x sec 2 f x
dx
53. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x iv y tan 5 x
cos 2 x dy sin x dy
5 tan 4 x sec 2 x
sec 2 x dx cos x dx
tan x
y tan f x v y cos e x
dy
f x sec 2 f x
dx
54. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x iv y tan 5 x
cos 2 x dy sin x dy
5 tan 4 x sec 2 x
sec 2 x dx cos x dx
tan x
y tan f x v y cos e x
dy
f x sec 2 f x dy
dx e x sin e x
dx