Practice Find f '(x)

harpazo

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Given y = (sin x)^(x), find dy/dx.
 

MarkFL

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Consider the following:

\(\displaystyle f(x)=(g(x))^{h(x)}\)

Take the natural log of both sides:

\(\displaystyle \ln(f(x))=h(x)\ln(g(x))\)

Implicitly differentiate:

\(\displaystyle \frac{1}{f(x)}\d{f}{x}=\frac{h(x)}{g(x)}+\d{h}{x}\ln(g(x))\)

\(\displaystyle \d{f}{x}=f(x)\left(\frac{h(x)}{g(x)}+\d{h}{x}\ln(g(x))\right)\)

\(\displaystyle \d{f}{x}=(g(x))^{h(x)}\left(\frac{h(x)}{g(x)}+\d{h}{x}\ln(g(x))\right)\)

In the given problem, we have:

\(\displaystyle f(x)=y,\,g(x)=\sin(x),\,h(x)=x\)

And so:

\(\displaystyle \d{y}{x}=\left(\sin(x)\right)^{x}\left(\frac{x}{\sin(x)}+\ln(\sin(x))\right)\)
 
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harpazo

Pure Mathematics
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I so look forward to our calculus trip.