Calculus I · Derivatives, rules, and implicit differentiation
Differentiate a power with two changing parts
Problem
For $x>0$, let $y=x^{\sin x}$. Derive $dy/dx$ by logarithmic differentiation and find the exact tangent line at $x=1$. Display the product-rule step, and state why neither a constant-power nor constant-base rule alone is sufficient.
Hint
Take logs: $\ln y=\sin x\ln x$.
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