find the difference quotient (f(x + h)-f(x))/...
find the difference quotient (f(x + h)-f(x))/h, where h ≠ 0, for the function below. f(x)=3x² + 5 simplify your answer as much as possible. (f(x + h)-f(x))/h =
Answer
# Explanation:
## Step1: Find $f(x + h)$
Substitute $x+h$ into $f(x)=3x^{2}+5$. So $f(x + h)=3(x + h)^{2}+5=3(x^{2}+2xh+h^{2})+5=3x^{2}+6xh + 3h^{2}+5$.
## Step2: Calculate $f(x + h)-f(x)$
$f(x + h)-f(x)=(3x^{2}+6xh + 3h^{2}+5)-(3x^{2}+5)=6xh+3h^{2}$.
## Step3: Find the difference - quotient
$\frac{f(x + h)-f(x)}{h}=\frac{6xh + 3h^{2}}{h}$. Since $h\neq0$, we can factor out $h$ from the numerator and cancel it with the denominator. So $\frac{h(6x + 3h)}{h}=6x+3h$.
# Answer:
$6x + 3h$