Question

What is the derivative of `tan(4x)^tan(5x)`?

Answer 1

We have the function

`y=tan(4x)^tan(5x)`

Take the natural logarithm of both sides:

`ln(y)=ln(tan(4x)^tan(5x))`

Using the rule `ln(a^b)=bln(a)`, rewrite the right hand side:

`ln(y)=tan(5x)*ln(tan(4x))`

Differentiate both sides. The chain rule will be in effect on the left hand side, and primarily we will use the product rule on the right hand side.

`dy/dx(1/y)=ln(tan(4x))d/dxtan(5x)+tan(5x)d/dxln(tan(4x))`

Differentiate each, again using the chain rule.

`dy/dx(1/y)=5sec^2(5x)ln(tan(4x))+tan(5x)((4sec^2(4x))/tan(4x))`

Multiply this all by `y`, which equals `tan(4x)^tan(5x)`, to solve for `dy/dx`.

`dy/dx=tan(4x)^tan(5x)(5sec^2(5x)ln(tan(4x))+(4sec^2(4x)tan(5x))/tan(4x))`