How to Find the Derivative of f(x)^g(x): Function Raised to a Function

How to find the derivative of y = (x)^g(x)
How to find the derivative of y = (x)^g(x)

In differential calculus, you learn rules for finding the derivative of the sum, difference, product, quotient, and composition of two functions. But what about a function raised to the power of another function? A function of the form y = f(x)^g(x) may seem tricky to derive, however, there is a special technique called the logarithmic transformation that allows you to find the first, second, third derivatives, etc.

Given a function y of the form y = f(x)^g(x), you first take the natural logarithm of both sides. This gives you the equivalent expression

Ln(y) = Ln[f(x)^g(x)]
Ln(y) = g(x)*Ln(f(x))

The last expression is true because log(a^b) = b*log(a), one of the fundamental rules of logarithms. If we take the derivative of both sides of the last expression, we get

y'/y = g'(x)*Ln(f(x)) + g(x)*(1/f(x))*f'(x)
y'/y = g'(x)*Ln(f(x)) + g(x)*f'(x)/f(x)
y' = y[ g'(x)*Ln(f(x)) + g(x)*f'(x)/f(x) ]
y' = [ f(x)^g(x) ]*[ g'(x)*Ln(f(x)) + g(x)*f'(x)/f(x) ]

Here are some examples of this derivation trick in action.

Example 1: Find the Derivative of y = x^x

One of the simplest examples of such a function is y = x^x. Using the logarithmic derivative technique, we have

y = x^x
Ln(y) = x*Ln(x)
y'/y = Ln(x) + 1
y' = y[Ln(x) + 1]
y' = (x^x)[Ln(x) + 1]

If you want to find the critical points of this function on the positive real axis, you set y' equal to 0 and solve for x. This gives you

(x^x)[Ln(x) + 1] = 0
Ln(x) + 1 = 0
Ln(x) = -1
x = 1/e.

The point x = 1/e is the global minimum of y = x^x over the positive real numbers.

Example 2: Find the Derivative of y = cos(x)^sin(x)

For this function, it is useful to know that the derivative of cos(x) is -sin(x), and that the derivative of sin(x) is cos(x).

y = cos(x)^sin(x)
Ln(y) = sin(x)Ln(cos(x))
y'/y = cos(x)Ln(cos(x)) + sin(x)*(-sin(x)/cos(x))
y'/y = cos(x)Ln(cos(x)) - (sin(x)^2)/cos(x)
y' = [cos(x)^sin(x)]*[cos(x)Ln(cos(x)) - (sin(x)^2)/cos(x)]

More by this Author

Comments 1 comment

Kyle 9 days ago

I never realized how difficult this was.

    Sign in or sign up and post using a HubPages Network account.

    0 of 8192 characters used
    Post Comment

    No HTML is allowed in comments, but URLs will be hyperlinked. Comments are not for promoting your articles or other sites.

    Click to Rate This Article