# Math Help - Derivative Of cos(x)^ln(x) ?

1. ## Derivative Of cos(x)^ln(x) ?

What is the derivative of cos(x)^ln(x)? When I plug it into mathematica, I don't know where the answer comes from...

2. Originally Posted by soma
What is the derivative of cos(x)^ln(x)? When I plug it into mathematica, I don't know where the answer comes from...
I would approach this by taking the logarithm of both sides.

$ln(y)=ln(cos^{ln(x)}x)$

$ln(y)=ln(x)lncos(x)$

The use implicit differentiation.

3. So I get e^((ln(cos(x))/x)-ln(x)tan(x))
But mathematica shows (cos(x)^ln(x))*((ln(cos(x))/x)-ln(x)tan(x))

The difference is I have e^, while mathematica has (cos(x)^ln(x))*

4. I get:

$\frac{1}{y}\frac{dy}{dx} =\frac{1}{x}lncos(x) -ln(x)tan(x)$

$\frac{dy}{dx}=\frac{y}{x}lncos(x)-yln(x)tan(x)$

5. Oh! don't forget that y= $cos^{ln(x)}(x)$

$\frac{dy}{dx}=cos^{ln(x)}(x)[\frac{lncos(x)}{x}-ln(x)tan(x)]$