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Calculus

Product Rule

The product rule is used to differentiate functions of the form \( f(x) = g(x)h(x) \), where \( g(x) \) and \( h(x) \) are two separate functions. The derivative of these types of functions is of the form \( f'(x) = g'(x)h(x) + g(x)h'(x) \).

Consider \( f(x) = (x^2 +2)(x^3) \):

If \( g(x) = x^2 + 2 \)

                                                                              \( \Longrightarrow g'(x) = 2x \)

If \( h(x) = x^3 \)

                                                                            \( \Longrightarrow h'(x) = 3x^2 \)

 

It follows that \( f'(x) = (2x)(x^3) + (x^2 + 2)(3x^2 ) \)

                                                            \( \Longrightarrow f'(x) = 2x^4 + 3x^4 + 6x^2 \)

                                                            \( \Longrightarrow f'(x) = 5x^4 + 6x^2 \)

Depending on the function you are dealing with, it may be easier to expand first if there are not too many terms in question. 

What if there is more than one rule to be applied? 

How would you differentiate \( f(x) = 2(x+5)^2 + x^3(x^2 -1)^3 \) ? 

See the video in the chain rule section for the solution. 

Quotient Rule

The quotient rule is used to differentiate functions of the form \( f(x) = \frac{g(x)}{h(x)} \), where \( g(x) \) and \( h(x) \) are separate functions.

The derivative of these types of functions is of the form \( f'(x) = \frac{g'(x)h(x) - h'(x)g(x)}{[h(x)]^2} \). 

Consider \( f(x) = \frac{x+2}{x^2} \)

If \( g(x) = x + 2 \)

                                                                               \( \Longrightarrow g'(x) = 1 \)

If \( h(x) = x^2 \)

                                                                                  \( \Longrightarrow h'(x) = 2x \)

 

It follows that \( f'(x) = \frac{1(x^2) - 2x(x+2)}{[x^2]^2} \) 

                                                                               \( \Longrightarrow f'(x) = \frac{x^2 - 2x^2 + 4x}{x^4} \) 

                                                                                                          \( \Longrightarrow f'(x) = \frac{-x^2 + 4x}{x^4} \)

Chain Rule

The chain rule is used to differentiate functions inside of functions. Sometimes applying the basic differentiation rules is not enough, and so we apply the chain rule. If \( f(x) = g(h(x)) \) where \( h(x) \) is some function, then \( f'(x) = g'(h(x))(h'(x)) \).

Recall \( f(x) = 2(x+5)^2 + x^3(x^2 -1)^3 \). How would you differentiate this function? 

See the video below for the solution:

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