Find the derivative of (sin x)cos x + (cos x)sin x

Find the derivative of the function

$f(x) = (\sin x)^{\cos x} + (\cos x)^{\sin x}.$

Rewrite each of the exponentials in the expression for $f(x)$ using the definition of the exponential function,

$f(x) = e^{\cos x \log (\sin x)} + e^{\sin x \log (\cos x)}.$

Now, we take the derivative directly using the chain rule and the formula for the derivative of the exponential function

$\begin{align*} f'(x) &= \left( e^{\cos x \log (\sin x)} \right) \left( \cos x \log (\sin x) \right)' \\ & \qquad + \left(e^{\sin x \log (\cos x)} \right) \left( \sin x \log (\cos x) \right)' \\[9pt] &= (\sin x)^{\cos x} \left( - \sin x \log (\sin x) + \frac{(\cos x)^2}{\sin x} \right) \\ & \qquad + (\cos x)^{\sin x} \left( \cos x \log (\cos x) - \frac{(\sin x)^2}{\cos x} \right) \\[9pt] &= (\sin x)^{1 + \cos x} \left( \cot^2 x - \log (\sin x)\right) - (\cos x)^{1+\sin x} \left( \tan^2 x - \log (\cos x) \right). \end{align*}$

Stumbling Robot

Find the derivative of (sin x)^{cos x} + (cos x)^{sin x}

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