A function g is defined for all real numbers and has the following property: g(a+b)-g(a)=4ab+2b^2. Find g'(x).

Answer 1

We proceed from the definition of the derivative:

g'(x) = [h→0]lim (g(x+h)-g(x))/h

Using the given property:

g'(x) = [h→0]lim (4xh+2h²)/h

Simplifying:

g'(x) = [h→0]lim 4x+2h

Evaluating:

g'(x) = 4x

And we are done.

Answer 2

Use this relation

g'(a) = [ g(a+b) - g(a) ] / b
= 4a + 2b

In the limit as b goes to zero, g'(a) = 4a
So g'(x) = 4x