Question

Calculats the derivative of the function. g(x)=(x^(2)-7x-1)^(-9)

Answer 1

`(-9(2x -7))/((x^2 - 7x - 1)^(10))`

Explanation:

1. To derive a function we use the power rule, it states:

The derivative of
`x^n` is
`nx^(n-1)`

2. Furthermore since this function has x in brackets to a power we need to use the chain rule

`(dy)/(dx)` =
`(du)/(dx)` x
`(dy)/(du)`

3. For this equation let

u =
`x^2 - 7x - 1`

y =
`u^(-9)`

lets differentiate u first using the power rule

`(du)/(dx)` = 2x -7

`(dy)/(du)` =
`-9u^{-10`

a negative exponent is simply
`(1)/(x^n)` therefore

`-9u^{-10` =
`(-9)/(u^(10))`

now we do
`(du)/(dx)` x
`(dy)/(du)`

`(-9)/(u^(10))` x 2x -7 =

`(-9(2x -7))/(u^(10))`

3. Now we replace the u with its original value

`(-9(2x -7))/((x^2 - 7x - 1)^(10))`