Find and simplify the difference quotient f(x+h)-f(x)/h, h cannot equal 0 for the given function. f(x) = 2x^2

Answer 1

(2(x+h)^2 - 2x^2)/h

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

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

= 4x+2h

at this point the limit of the original difference quotient = 4x as h goes to zero. But you couldn't let h=0 until you got h eliminated as the denominator.

4x is the derivative of f(x) = 2x^2 or

f'(x)=4x. Multiply the exponent 2, by the coefficient 2, to get 4 and reduce the exponent by 1. that's the short cut way to take a derivative.

But the difference quotient is a longer method.