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

1 Answer
Apr 24, 2016

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))#