Question Statement

Find the derivative of $y=\sin x\cos x$ with respect to $x$.

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Background and Explanation

This problem involves differentiating a product of two trigonometric functions. You will need to apply the product rule for differentiation, along with standard trigonometric derivatives and the double angle identity for cosine to simplify the final result.

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Solution

We are given the function:

$y=\sin x\cos x$

To find $\frac{dy}{dx}$, we recognize this as a product of two functions where $u=\sin x$ and $v=\cos x$. We apply the product rule for differentiation:

$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$

Setting up the differentiation:

$\frac{dy}{dx}=\frac{d}{dx}(\sin x\cos x)$

Applying the product rule (differentiating the first function times the second, plus the first function times the derivative of the second):

$=\cos x\cdot \frac{d}{dx}(\sin x)+\sin x\cdot \frac{d}{dx}(\cos x)$

Substituting the standard derivatives $\frac{d}{dx}(\sin x)=\cos x$ and $\frac{d}{dx}(\cos x)=-\sin x$:

$=\cos x(\cos x)+\sin x(-\sin x)$

Performing the multiplication:

$={\cos }^{2}x-{\sin }^{2}x$

Recognizing this as the double angle identity for cosine:

$=\cos 2x$

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Key Formulas or Methods Used

• Product Rule: $\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$
• Trigonometric Derivatives: $\frac{d}{dx}(\sin x)=\cos x$ and $\frac{d}{dx}(\cos x)=-\sin x$
• Double Angle Identity: $\cos 2x={\cos }^{2}x-{\sin }^{2}x$

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Summary of Steps

1. Identify $y=\sin x\cos x$ as a product of two trigonometric functions
2. Apply the product rule: $\frac{dy}{dx}=\cos x\cdot \frac{d}{dx}(\sin x)+\sin x\cdot \frac{d}{dx}(\cos x)$
3. Substitute the derivatives: $\cos x(\cos x)+\sin x(-\sin x)$
4. Simplify the expression to ${\cos }^{2}x-{\sin }^{2}x$
5. Apply the double angle identity to obtain the final answer: $\cos 2x$