3. (5 points) let f(x)=2x² - 4x + 6. find the derivative f(x) by using the definition of the derivative of f as a limit of a difference quotient, and use algebraic calculations to compute the limit.

Question

Accepted Answer

# Explanation:
## Step1: Recall the definition of the derivative
The definition of the derivative of a function $y = f(x)$ is $f'(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}$. Given $f(x)=2x^{2}-4x + 6$, first find $f(x + h)$.
$$
\begin{align*}
f(x + h)&=2(x + h)^{2}-4(x + h)+6\
&=2(x^{2}+2xh+h^{2})-4x-4h + 6\
&=2x^{2}+4xh+2h^{2}-4x-4h + 6
\end{align*}
$$
## Step2: Calculate $f(x + h)-f(x)$
$$
\begin{align*}
f(x + h)-f(x)&=(2x^{2}+4xh+2h^{2}-4x-4h + 6)-(2x^{2}-4x + 6)\
&=2x^{2}+4xh+2h^{2}-4x-4h + 6-2x^{2}+4x - 6\
&=4xh+2h^{2}-4h
\end{align*}
$$
## Step3: Calculate $\frac{f(x + h)-f(x)}{h}$
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{4xh+2h^{2}-4h}{h}\
&=\frac{h(4x + 2h-4)}{h}\
&=4x+2h - 4
\end{align*}
$$
## Step4: Compute the limit as $h
ightarrow0$
$$
\begin{align*}
f'(x)&=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}\
&=\lim_{h
ightarrow0}(4x+2h - 4)\
&=4x-4
\end{align*}
$$
# Answer:
$f'(x)=4x - 4$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the definition of the derivative

Step2: Calculate $f(x + h)-f(x)$

Step3: Calculate $\frac{f(x + h)-f(x)}{h}$

Step4: Compute the limit as $h

Answer: