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

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Answer 1 · 2016-04-24T13:35:48

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 $dy/dx$ .

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

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