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

Explanation:

Step1: Substitute (f(z)) and (f(x))

Given (f(x)=2x^{2}-5x + 5), then (f(z)=2z^{2}-5z + 5). [ \begin{align*} \frac{f(z)-f(x)}{z - x}&=\frac{(2z^{2}-5z + 5)-(2x^{2}-5x + 5)}{z - x}\ &=\frac{2z^{2}-5z+5 - 2x^{2}+5x - 5}{z - x}\ &=\frac{2(z^{2}-x^{2})-5(z - x)}{z - x} \end{align*} ]

Step2: Factor and simplify

Use the formula (a^{2}-b^{2}=(a + b)(a - b)), so (z^{2}-x^{2}=(z + x)(z - x)). [ \begin{align*} \frac{2(z^{2}-x^{2})-5(z - x)}{z - x}&=\frac{2(z + x)(z - x)-5(z - x)}{z - x}\ &=\frac{(z - x)[2(z + x)-5]}{z - x}\ &=2(z + x)-5 \end{align*} ]

Step3: Find the limit

Now, find (\lim_{z\rightarrow x}(2(z + x)-5)). Substitute (z=x) into (2(z + x)-5). [ \begin{align*} \lim_{z\rightarrow x}(2(z + x)-5)&=2(x + x)-5\ &=4x-5 \end{align*} ]

Answer:

(4x - 5)