find the difference quotient of f; that is, find (f(x + h)-f(x))/h, h≠0, for the following function. be sure…

Explanation:

Step1: Find f(x + h)

Substitute x + h into f(x): $f(x + h)=\frac{10}{(x + h)^2}$

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

$f(x + h)-f(x)=\frac{10}{(x + h)^2}-\frac{10}{x^2}=\frac{10x^2-10(x + h)^2}{x^2(x + h)^2}$ Expand $(x + h)^2=x^2 + 2xh+h^2$: $f(x + h)-f(x)=\frac{10x^2-10(x^2 + 2xh+h^2)}{x^2(x + h)^2}=\frac{10x^2-10x^2-20xh - 10h^2}{x^2(x + h)^2}=\frac{-20xh-10h^2}{x^2(x + h)^2}$

Step3: Calculate the difference - quotient

$\frac{f(x + h)-f(x)}{h}=\frac{\frac{-20xh - 10h^2}{x^2(x + h)^2}}{h}=\frac{-20xh-10h^2}{x^2(x + h)^2}\cdot\frac{1}{h}=\frac{-10h(2x + h)}{x^2(x + h)^2}\cdot\frac{1}{h}=\frac{-10(2x + h)}{x^2(x + h)^2}$

Answer:

$\frac{-10(2x + h)}{x^2(x + h)^2}$