What is the derivative of $x^(cosx)$ ?

Question

1696 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-07-29T18:08:54

We shall attack this problem by logarithmic differentiation.

Let, $y = x^(cos x)$

Taking natural logarithm on both the sides, we obtain

$Ln y = Cos x Ln x$

Now, differentiating with respect to $x$ ,

$1/y(dy)/dx = Cos x/x - Sin xLn x$

In the last step, I used chain rule on the LHS and the product rule on the RHS.

Thus, $(dy)/dx = x^Cos x [Cos x/x - Sin xLn x]$

What is the derivative of x^(cosx)x^(cosx)?