Question

find the difference quotient $\frac{f(x + h)-f(x)}{h}$, where $h\
eq0$, for the function below.
$f(x)=5x^{2}+7$
simplify your answer as much as possible.
$\frac{f(x + h)-f(x)}{h}=\square$

Explanation:

Step1: Find \( f(x + h) \)

Substitute \( x+h \) into \( f(x)=5x^{2}+7 \).
\( f(x + h)=5(x + h)^{2}+7 \).
Expand \( (x + h)^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), so \( f(x + h)=5(x^{2}+2xh+h^{2})+7=5x^{2}+10xh + 5h^{2}+7 \).

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

\( f(x + h)-f(x)=(5x^{2}+10xh + 5h^{2}+7)-(5x^{2}+7) \).
Remove the parentheses: \( f(x + h)-f(x)=5x^{2}+10xh + 5h^{2}+7 - 5x^{2}-7 \).
Simplify by combining like - terms: \( f(x + h)-f(x)=10xh+5h^{2} \).

Step3: Calculate the difference quotient \( \frac{f(x + h)-f(x)}{h} \)

Substitute \( f(x + h)-f(x)=10xh + 5h^{2} \) into \( \frac{f(x + h)-f(x)}{h} \).
\( \frac{f(x + h)-f(x)}{h}=\frac{10xh+5h^{2}}{h} \).
Factor out \( h \) from the numerator: \( \frac{f(x + h)-f(x)}{h}=\frac{h(10x + 5h)}{h} \).
Since \( h
eq0 \), cancel out the \( h \) terms.

Answer:

\(10x + 5h\)