The document outlines different topics related to calculus derivatives including definitions of derivatives, the power rule for raising terms to a power, the product rule for multiplying terms, and the quotient rule for dividing terms. It provides examples of functions that can use each rule, such as applying the power rule to raise a term with an exponent like x^2 to derivative 2x, or using the product rule to derive the derivative of two multiplied terms. The document aims to introduce these fundamental calculus derivative rules and concepts at a basic level.

2. Outline

 These will be some of the topics that I will be
covering in this website.
 This is just a simplified version of each one of these
topics that I will be covering more in depth In a
separate post.

3. Learning about derivatives

 definition of a derivative

4. Power rule

 When your x term has a power you simply bring it
to the front and then you subtract one from the
original power to gain your new power.

5. Functions that you can use the
power rule with

 f(x) = x² + 10

2x
 f(x) = 3x ² + 5x

6x + 5
 f(x) = 4x3 + 2x2 + x

12x2 + 4x + 1

6. Product rule

 The derivative of a function where two terms are
being multiplied can be expressed as the function of
the first term times the derivative of the next term
plus the derivative of the first term times the
function of the second term.

7. Functions that you can use the
Product rule with

f(x) = 2x + 3
g(x) = 4x + 5
(f ⋅ g)′ = ( 2x + 3)( 4) + (2)(4x +5)
= (8x + 12) + (8x +10 )
= 16x + 22

8. Quotient rule

 The derivative of a function with terms that are
being divided can be expressed by the function of
the denominator times the derivative of the
numerator minus the numerator times the derivative
of the denominator, all over the denominator
squared.

9. Functions that you can use the
quotient rule with

 ((3x + 5) /(4x+ 2)) ′ = ((4x + 2)(3) – (3x + 5)(4))/(4x + 2)2
 = 12x + 6 – (12x + 20) /(4x + 2)2
 = -14/(4x + 2)2

10. Calculus is fun
