Question

How do you find the derivative of `x^(sinx)`?

Answer 1

`d/dx x^sinx= x^sinx((sinx)/x -cosxlnx)`

Explanation:

Let `y = x^sinx`

Then `ln y = ln(x^sinx)`

`:. lny = (sinx)lnx`

Differentiating implicitly and applying the product rule:

`1/ydy/dx=(sinx)(1/x) + (-cosx)(lnx)`
`1/ydy/dx=(sinx)/x -cosxlnx`
`dy/dx=y((sinx)/x -cosxlnx)`
`dy/dx=x^sinx((sinx)/x -cosxlnx)`