Question

Question #e4f55

Answer 1

Use logarithmic differentiation to obtain `d/dx[(sin(x))^(cos(x))]`

`=(sin(x))^(cos(x)) * (cos(x)cot(x)-ln(sin(x))sin(x))`.

Explanation:

Let `y=(sin(x))^(cos(x))`. Then `ln(y)=cos(x) * ln(sin(x))` (since `ln(a^b)=b*ln(a)`).

Now differentiate both sides of this equation with respect to `x`, keeping in mind that `y` is a function of `x` to get

`1/y * dy/dx = -sin(x) * ln(sin(x)) + cos(x) * 1/sin(x)*cos(x)`.

Multiplying both sides by `y=(sin(x))^(cos(x))` and rearranging gives the answer

`dy/dx = d/dx[(sin(x))^(cos(x))]`

`=(sin(x))^(cos(x)) * (cos(x)cot(x)-ln(sin(x))sin(x))`.