Find the Derivative of the function?

Question

Find the derivative of the function.

f(x) = 5/(x^2 − 9)^4

Answer 1

f(x) = 5/[x^2 -9]^4

quotient rule is the derivative = the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared

derivative of the denominator = 4(x^2-9)^3 times the derivative of x^2 = 4(x^2-9)^3(2), by the chain rule

f'(x) = -5(4)(x^2-9)^3(2)/(x2-9)^8

=--40(x^2-9)^-5

= -40/(x^2-9)^5

Answer 2

To find d/dx[f(x)] where f(x)= 5/[(x² -9)⁴] =5.(x² - 9)^-4

Using the chain rule we get:

d/dx[f(x)] = 5. -4. (x² -9)^-5 .2x = -40x(x² -9)^-5 Ans.

Answer 3

This can be approached with a combination of the quotient rule/chain rule, or with the chain rule directly (this is quicker).

Note that

f(x) = 5 ( x^2-9 )^(-4)

By the chain rule,

f'(x) = 5 (-4) ( x^2-9 )^(-5) (2x) = -40x( x^2-9)^(-5) = -40x / ( x^2 -9 )^5