Calculus Function

Question

For the function, find and simplify

f ( x + h) - f ( x ) ÷ h

Assume h≠0

f(x) = 4 ÷ x²

Answer 1

f'(x) = -8/x^3

f(x) = 4/x^2

f(x+h) = 4/(x+h)^2 = 4/(x^2+2xh+h^2)

[f(x+h) - f(x)]/h = [4/(x^2+2xh+h^2) - 4/x^2]/h combine over a common denominator

= [4x^2 -4x^2 -8xh -4h^2]/(x^2)(x^2+2xh+h^2)x^2h = (-8x -4h)/(x^2+2xh+h^2)x^2h

as h approaches 0, [f(x+h)-f(x)]/h approaches -8x/x^4=-8/x^3

the limit of the difference quotient = the derivative of f(x)