use limits to compute the derivative. f(9) where f(x)=x^2 - 2 f(9)= (simplify your answer.)
use limits to compute the derivative. f(9) where f(x)=x^2 - 2 f(9)= (simplify your answer.)
Answer
Explanation:
Step1: Recall derivative limit - definition
The derivative of a function $f(x)$ at a point $a$ is given by $f^{\prime}(a)=\lim_{h\rightarrow0}\frac{f(a + h)-f(a)}{h}$. Here, $a = 9$ and $f(x)=x^{2}-2$.
Step2: Find $f(9 + h)$ and $f(9)$
$f(9 + h)=(9 + h)^{2}-2=81+18h+h^{2}-2=h^{2}+18h + 79$. $f(9)=9^{2}-2=81 - 2=79$.
Step3: Substitute into the limit - formula
$f^{\prime}(9)=\lim_{h\rightarrow0}\frac{f(9 + h)-f(9)}{h}=\lim_{h\rightarrow0}\frac{(h^{2}+18h + 79)-79}{h}$.
Step4: Simplify the expression
$\lim_{h\rightarrow0}\frac{h^{2}+18h + 79 - 79}{h}=\lim_{h\rightarrow0}\frac{h^{2}+18h}{h}=\lim_{h\rightarrow0}(h + 18)$.
Step5: Evaluate the limit
As $h\rightarrow0$, $\lim_{h\rightarrow0}(h + 18)=18$.
Answer:
$18$