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Chapter 6: Problem 20

If \(y=(\sin x)^{x}\), find \(y=(\sin x)^{x}\).

Short Answer

Expert verified
The derivative of the function \(y=(\sin x)^{x}\) is \(y'= (\sin x)^{x}[ln(\sin x) + cot(x)]\).

Step by step solution

01

Apply the logarithmic differentiation

To simplify the differentiation process for this function, apply logarithmic differentiation. This means taking the natural log of both sides. Write \(ln(y)=x \cdot ln(\sin x)\).
02

Differentiate both sides

Now, differentiate both sides of equation with respect to \(x\). The derivative of \(ln(y)\) is \(1/y \cdot y'\) by the chain rule, and the right side requires product rule and chain rule, which gives \(x' \cdot ln(\sin x)\) + \(x \cdot \frac{cos(x)}{sin(x)}\). Simplifying, we get \(1/y \cdot y'= ln(\sin x) + cot(x)\).
03

Isolate \(y'\)

Multiply both sides by \(y\) to isolate \(y'\) on one side of the equation, we get \(y'=y[ln(\sin x) + cot(x)]\).
04

Substitute back for \(y\)

Since we have \(y=(\sin x)^{x}\), substitute it back in, we obtain \(y'= (\sin x)^{x}[ln(\sin x) + cot(x)]\)

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