Question #e4f55

1 Answer
Dec 27, 2017

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)).