Question

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

Answer 1

When you're attempting to differentiate a function involving a variable raised to an expression also involving that variable, it's a good sign that you'll need to use logarithmic differentiation.

Logarithmic differentiation involves taking the natural log of both sides of the equation before differentiating. This way, we can use the exponents property of logs to essentially turn the exponent into a multiplier.

In this case, we have:

`ln y = ln (x^(tan x))`

Bring the `tan x` out front:

`ln y = tan x ln x`

Now this is very simple to differentiate, so long as one is vaguely familiar with implicit differentiation. We will apply the product rule to the right-hand side of the equation, and we will apply the chain rule to the left:

`1/y y' = sec^2 x ln x + tan x/x`

Multiply both sides by `y`:

`y' = y(sec^2 x ln x + tan x/x)`

Now simply substitute for `y`:

`y' = x^tan x (sec^2 x ln x + tan x/x)`

And there's our derivative.