use the formula f(x)=\lim_{z\to x}\frac{f(z)-f(x)}{z - x} to find the derivative of the following function. f(x)=2x^{2}-x + 4 f(x)=\square

Question

Accepted Answer

# Explanation:
## Step1: Find f(z)
Substitute z into f(x): $f(z)=2z^{2}-z + 4$
## Step2: Calculate f(z)-f(x)
$$
\begin{align*}
f(z)-f(x)&=(2z^{2}-z + 4)-(2x^{2}-x + 4)\
&=2z^{2}-z + 4-2x^{2}+x - 4\
&=2(z^{2}-x^{2})-(z - x)
\end{align*}
$$
## Step3: Simplify $\frac{f(z)-f(x)}{z - x}$
$$
\begin{align*}
\frac{f(z)-f(x)}{z - x}&=\frac{2(z^{2}-x^{2})-(z - x)}{z - x}\
&=\frac{2(z - x)(z + x)-(z - x)}{z - x}\
&=\frac{(z - x)[2(z + x)-1]}{z - x}\
&=2(z + x)-1
\end{align*}
$$
## Step4: Find the limit as z→x
$$
\begin{align*}
f^{\prime}(x)&=\lim_{z
ightarrow x}\frac{f(z)-f(x)}{z - x}\
&=\lim_{z
ightarrow x}(2(z + x)-1)\
&=2(x + x)-1\
&=4x-1
\end{align*}
$$
# Answer:
$4x - 1$

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QUESTION IMAGE

Question

Explanation:

Step1: Find f(z)

Step2: Calculate f(z)-f(x)

Step3: Simplify $\frac{f(z)-f(x)}{z - x}$

Step4: Find the limit as z→x

Answer: