Question

How can I find the derivative of f (x)= 3÷(x-2)^2?

Answer 1

`-\frac{6}{(x-2)^3}`

Explanation:

First of all, you can factor out constants:

`d/(dx) \frac{3}{(x-2)^2} = 3d/(dx) \frac{1}{(x-2)^2}= 3d/(dx) (x-2)^{-2}`

Now we use the rule

`d/(dx) f(x)^n = nf(x)^{n-1}f'(x)`

where `f(x)=x-2` and `n=-3`. This implies `n-1=-3` and `f'(x)=1`

Thus,

`d/(dx) (x-2)^{-2} = -2(x-2)^{-3}=-\frac{2}{(x-2)^3}`

Remember that there was a `3` multiplying the whole function to get

`3d/(dx) (x-2)^{-2} = -\frac{6}{(x-2)^3}`

Answer 2

`f'(x)=-6/(x-2)^3`

Explanation:

`"differentiate using the "color(blue)"chain rule"`

`"given "f(x)=g(h(x))" then"`

`f'(x)=g'(h(x))xxh'(x)larrcolor(blue)"chain rule"`

`f(x)=3/(x-2)^2=3(x-2)^(-2)`

`f'(x)=-6(x-2)^(-3)xxd/dx(x-2)`

`color(white)(f'(x))=-6/(x-2)^3`