Derivative of a Constant Multiple of a Function Examples

Let g(x) = 3x². What is g ' (x)?

The function g is the constant 3 multiplied by the function x². The derivative of g is 3 multiplied by the derivative of x²:

g ' (x) = 3(x²)'
 = 3(2x)
 = 6x.

Let f(x) = 4sin(x). What is f ' (x)?

It doesn't matter that we're using f instead of g for the name of the function; the idea is the same. Since f is the constant 4 multiplied by sin(x), the derivative of f is the constant 4 multiplied by the derivative of sin(x):

f ' (x) = 4(sin x)'
 = 4(cos x)
 = 4cos x.