use the formula f(x)=\\lim_{z\\to x}\\frac{f(z)-f(x)}{z - x} to find the derivative of the following…

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\rightarrow x}\frac{f(z)-f(x)}{z - x}\ &=\lim_{z\rightarrow x}(2(z + x)-1)\ &=2(x + x)-1\ &=4x-1 \end{align*} ]

Answer:

$4x - 1$