select the correct answer. if $f(x)=5x^{2}-3x + 2$, find $lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}$. \na…

Answer:

D. $10x - 3$

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=5(x + h)^2-3(x + h)+2\ &=5(x^{2}+2xh+h^{2})-3x-3h + 2\ &=5x^{2}+10xh+5h^{2}-3x-3h + 2 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(5x^{2}+10xh+5h^{2}-3x-3h + 2)-(5x^{2}-3x + 2)\ &=5x^{2}+10xh+5h^{2}-3x-3h + 2 - 5x^{2}+3x - 2\ &=10xh+5h^{2}-3h \end{align*} ]

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

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{10xh+5h^{2}-3h}{h}\ &=\frac{h(10x + 5h-3)}{h}\ &=10x+5h - 3 \end{align*} ]

Step4: Find the limit as $h\rightarrow0$

[ \begin{align*} \lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}&=\lim_{h\rightarrow0}(10x+5h - 3)\ &=10x-3 \end{align*} ]