Question

find a formula for the derivative of the function g(x)=5x^2 - 4 using the limit definition of the derivative: g(x)=lim_{h→0}(( )/h)=( ). (in the first answer blank, fill in the numerator of the difference quotient you use to evaluate the derivative. in the second, fill out the derivative you obtain after completing the limit calculation.)

Explanation:

Step1: Recall limit - definition of derivative

The limit - definition of the derivative of a function $y = g(x)$ is $g^{\prime}(x)=\lim_{h
ightarrow0}\frac{g(x + h)-g(x)}{h}$. First, find $g(x + h)$ for $g(x)=5x^{2}-4$.
$g(x + h)=5(x + h)^{2}-4=5(x^{2}+2xh+h^{2})-4=5x^{2}+10xh + 5h^{2}-4$.
Then $g(x + h)-g(x)=(5x^{2}+10xh + 5h^{2}-4)-(5x^{2}-4)=10xh+5h^{2}$.

Step2: Evaluate the limit

$g^{\prime}(x)=\lim_{h
ightarrow0}\frac{g(x + h)-g(x)}{h}=\lim_{h
ightarrow0}\frac{10xh + 5h^{2}}{h}$.
Factor out an $h$ from the numerator: $\lim_{h
ightarrow0}\frac{h(10x + 5h)}{h}$.
Cancel out the $h$ terms ($h
eq0$ as we are taking the limit as $h$ approaches 0, not setting $h = 0$): $\lim_{h
ightarrow0}(10x + 5h)$.
As $h
ightarrow0$, we get $10x$.

Answer:

First blank: $10xh+5h^{2}$
Second blank: $10x$