Question

Use the definition of the derivative to find f' for each function f.

$f\left(x\right)=-2x^2$

Short answer

$f\text{'}\left(x\right)=$$-4x$

Step-by-step solution

Step 1. Given information. 

We have been given a function:

$f\left(x\right)=-2x^2$

We have to find f' using the definition of the derivative.

Step 2. Find the derivative

Derivative of the function can be found as :

$\begin{array}{r}\underset{h\to 0}{lim} \frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{lim} \frac{-2(x+h{)}^2-\left(-2x^2\right)}{h}\\ =\underset{h\to 0}{lim} \frac{-2\left(x^2+2xh+h^2\right)+2x^2}{h}\\ =\underset{h\to 0}{lim} \frac{-4xh-2h^2}{h}\\ =\underset{h\to 0}{lim} \frac{h(-4x-2h)}{h}\\ =\underset{h\to 0}{lim} -4x-2h\\ =-4x\end{array}$