find the difference quotient of f(x)=x² + 5; that is find (f(x + h)-f(x))/h, h≠0. be sure to simplify. the…
find the difference quotient of f(x)=x² + 5; that is find (f(x + h)-f(x))/h, h≠0. be sure to simplify. the difference quotient is .
Answer
Explanation:
Step1: Find f(x + h)
Given (f(x)=x^{2}+5), then (f(x + h)=(x + h)^{2}+5=x^{2}+2xh+h^{2}+5)
Step2: Substitute f(x + h) and f(x) into the difference - quotient formula
(\frac{f(x + h)-f(x)}{h}=\frac{(x^{2}+2xh + h^{2}+5)-(x^{2}+5)}{h})
Step3: Simplify the numerator
((x^{2}+2xh + h^{2}+5)-(x^{2}+5)=x^{2}+2xh + h^{2}+5 - x^{2}-5=2xh+h^{2}) So, (\frac{f(x + h)-f(x)}{h}=\frac{2xh + h^{2}}{h})
Step4: Factor out h from the numerator and simplify
(\frac{2xh + h^{2}}{h}=\frac{h(2x + h)}{h}=2x+h)
Answer:
(2x + h)