Derivative of Function to Power of Function

Theorem
Let $$u \left({x}\right), v \left({x}\right)$$ be real functions which are differentiable on $$\reals$$.

Then:
 * $$D_x \left({u^v}\right) = v u^{v-1} D_x \left({u}\right) + u^v \ln u D_x \left({v}\right)$$.

Proof
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Note
When $$u = x$$ and $$v = n$$ where $$n$$ is constant, we get the Power Rule for Derivatives:
 * $$D_x \left({x^n}\right) = n x^{n-1}$$.

When $$v = x$$ and $$u = a$$ where $$a$$ is constant, we get the Derivative of Exponential Function:
 * $$D_x \left({a^x}\right) = a^x \ln a$$.