Question

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

Answer 1

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.