Derivatives of Constant Multiples of Functions

To take the derivative of a constant multiple, we have our usual two steps: first, recognize the constant multiple, which means finding the constant and the function it multiplies, and applying the rule for constant multiples.

Recognize the constant multiple: Let's consider a some examples -- say, g(x) = 5 e^x, h(x) = e tan(x), and p(x) = ln(a) ln(x). Are these constant multiples? Sure: for g(x), the constant 5 multiplies the function e^x; for h(x), the constant e multiplies the function tan(x); and for p(x), ln(a) is a constant because a isn't a variable, so we have the function ln(x) multiplied by the constant ln(a). Then we

Apply the Rule for Constant Multiples:

So to find the derivative of a constant multiple, we just take the derivative of the function and multiply by the constant. We show this for each of our example functions in the following table:

we start with		c		f(x)		f '(x)		final derivative
g(x) = 5 ex		5		ex		ex		g'(x) = 5 ex
h(x) = e tan(x)		e		tan(x)		1 / cos2(x)		h '(x) = e / cos2(x)
p(x) = ln(a) ln(x)		ln(a)		ln(x)		1 / x		p'(x) = ln(a) / x