Derivation
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1. This document discusses various techniques of differentiation including the sum rule, difference rule, power rule, product rule, and quotient rule. 2. The sum and difference rules state that the derivative of a sum or difference of functions is equal to the sum or difference of the individual derivatives. 3. The power rule states that the derivative of a variable raised to a power is equal to that power multiplied by the derivative of the variable raised to the power minus one.
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Integral and Differential CalculusI.pptx
The document provides information about differentiation and finding derivatives of functions:
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Group No 05, calculus.pptx
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Derivation
- 1. 1
- 3. 3 “Differentiation is a process of finding a rate of change of one variable with respect to other.” • A change in variable is represented by: 𝑑𝑦 𝑑𝑥 “ ”
- 4. Differentiation is the part of the science of the Calculus, which was firstly developed by • ISSAC NEWTON • GOTTFRIED LEIBNIZ HISTORY 4 “in 17th Century”
- 5. WHY WE USE ? • • • • • Maximum and Minimum values Of particular Fucntions: 5
- 7. “SUM RULE: 7 In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. The sum rule states that for two functions f(y) and g(y): 𝑑 𝑑𝑥 (f(y) + g(y))= 𝑑 𝑑𝑥 (f(y)) + 𝑑 𝑑𝑥 (g(y))
- 8. 8 EXAMPLE OF SUM RULE Find the Derivative of y=12x+36 using the sum rule. y=12x+36 Differentiate w.r.t “x” into both sides 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 (12x + 36) 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 12x + 𝑑 𝑑𝑥 (36) 𝒅𝒚 𝒅𝒙 = 𝟏𝟐 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 12 . 1
- 9. “DIFFERENCE RULE: 9 In calculus, the Difference rule in differentiation is a method of finding the derivative of a function that is the difference of two other functions for which derivatives exist. The sum rule states that for two functions f(y) and g(y): 𝑑 𝑑𝑥 (f(y) − g(y))= 𝑑 𝑑𝑥 (f(y)) − 𝑑 𝑑𝑥 (g(y))
- 10. 10 EXAMPLE OF DIFFERENCE RULE Find the Derivative of y=-12x-104 using the Difference rule. y=34x-12x-104 Differentiate w.r.t “x” into both sides 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 (-12x-104) 𝑑 𝑑𝑥 (y)=- 𝑑 𝑑𝑥 (12x) - 𝑑 𝑑𝑥 (104) 𝒅𝒚 𝒅𝒙 = -12
- 11. “POWER RULE: 11 “The power rule in calculus is a simple rule that helps you find the derivative of a variable raised to a power.” The power rule of an n-integer states: “ 𝑑 𝑑𝑥 𝑥 𝑛 =n. 𝑑 𝑑𝑥 𝑥 𝑛−1 ”
- 12. 12 EXAMPLE OF POWER RULE Find the derivative of y=𝑥5 -3𝑥2 +9x using the power rule. y=𝒙 𝟓 -3𝒙 𝟐 +9x Differentiate w.r.t “x” into both sides 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 (𝑥5 -3𝑥2 +9x) 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 (𝑥5 )- 𝑑 𝑑𝑥 (3𝑥2 ) + 𝑑 𝑑𝑥 (9x) 𝒅𝒚 𝒅𝒙 = 𝟓𝒙 𝟒 − 𝟔x + 9
- 13. “PRODUCT RULE: 13 In Calculus, the Product rule is a formula used to find the derivatives of products of two or more function. It may be stated as: The sum rule states that for two functions f(y) and g(y): 𝑑 𝑑𝑥 (f(y).g(y))=f(y). 𝑑 𝑑𝑥 g(y) + g(y). 𝑑 𝑑𝑥 f(y)
- 14. 14 EXAMPLE OF PRODUCT RULE Find the derivative of y=𝑥2 𝑙𝑛𝑥 using the product rule. y=𝒙 𝟐 𝒍𝒏𝒙 Differentiate w.r.t “x” into both sides 𝑑 𝑑𝑥 (y)=𝑥2 . 𝑑 𝑑𝑥 (lnx)+lnx 𝑑 𝑑𝑥 (𝑥2 ) 𝑑 𝑑𝑥 (y)=𝑥2 . 1 𝑥 + lnx.2x dy dx = x + 2xlnx 𝒅𝒚 𝒅𝒙 = 𝒙(𝟏 + 𝟐𝒍𝒏𝒙)
- 15. “QUOTIENT RULE: 15 “In Calculus, the Quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.” Let 𝑥 𝑦 be the function. 𝑑𝑦 𝑑𝑥 = 𝑑𝑥 𝑑𝑦 . 𝑦 − 𝑑𝑦 𝑑𝑥 . 𝑥 𝑦2 “ ”
- 16. 16 EXAMPLE OF QUOTIENT RULE Find the derivative of y= 𝟐𝒙+𝟓 𝟑𝒙−𝟐 using the quotient Rule..? y= 𝟐𝒙+𝟓 𝟑𝒙−𝟐 Differentiate w.r.t “x” into both sides 𝑑 𝑑𝑥 (y)= 2𝑥+5. 𝑑𝑦 𝑑𝑥 3𝑥−2 − 3𝑥−2 𝑑𝑦 𝑑𝑥 (2𝑥+5) (3𝑥−2)2 𝑑 𝑑𝑥 (y)= 2𝑥+5 . 3 − 3𝑥−2 .(2) (3𝑥−2)2
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