Question

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

Answer 1

We use a technique called logarithmic differentiation to differentiate this kind of function.

In short, we let `y = (cos(x))^x`,

Then,

`ln(y) = ln((cos(x))^x)`

`ln(y) = xln(cos(x))`, by law of logarithms,

And now we differentiate.

`d/dx(ln(y)) = d/dx(xln(cos(x)))`

`dy/dx xx d/dy(ln(y)) = ln(cos(x)) xx d/dx(x) + x d/dx(ln(cos(x)))`

`dy/dx xx 1/y = ln(cos(x)) - (xsin(x))/cos(x)`

`dy/dx = y(ln(cos(x)) - (xsin(x))/cos(x))`

`dy/dx = (cos(x))^x(ln(cos(x)) - (xsin(x))/cos(x))`

Alternatively, you can express `(cos(x))^x` as `e^(xln(cos(x)))`, but that's basically the same thing.