Question

find the differnece quotient for g(x)=4x^3

Answer 1

g(x)=4x³; g(x+h)=4(x+h)³=4(x³+3x²h+3xh²+h³).

g(x+h)-g(x)=4(x³+3x²h+3xh²+h³)-4x³=12x²h+12xh²+4h³.

The difference quotient is (12x²h+12xh²+4h³)/((x+h)-x)=(12x²h+12xh²+4h³)/h=12x²+12xh+4h².

As h→0 this becomes 12x² and is called the derivative of 4x³.

It can also be called the gradient or slope or differential.