How do you find the derivative of y=x^tan(x)?

1 Answer
Aug 1, 2014

This is a type of problem involving logarithmic differentiation.

Whenever you're trying to differentiate a variable raised to some power also involving that variable, it's a good hint that logarithmic differentiation will help you out.

1.) y = x^tanx

The first step is to take the natural log of both sides:

2.) ln y = ln x^tanx

Using the exponents property of logarithms, we bring the exponent out in front of the log as a multiplier. This is done to make differentiating easier:

3.) ln y = tan x * ln x

Now we implicitly differentiate, taking care to use the chain rule on ln y. We will also apply the product rule to the right side of the equation:

4.) 1/y * dy/dx = d/dx[tan x] * ln x + d/dx[ln x] * tan x

We know that the derivative of tan x is equal to sec^2 x, and the derivative of ln x is 1/x:

5.) 1/y * dy/dx = sec^2 x ln x + tan x / x

Multiply both sides by y to isolate dy/dx:

6.) dy/dx = y(sec^2 x ln x + tan x / x)

We know y from step 1, so we will substitute:

7.) dy/dx = x^tan x (sec^2 x ln x + tan x / x)

And there is the derivative.