Question Statement

Find the derivative of the function $y=3\cos x-5\cot x$ with respect to $x$.

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Background and Explanation

This problem requires applying basic differentiation rules to trigonometric functions. You need to know the standard derivatives of cosine and cotangent, along with the constant multiple rule and difference rule for differentiation.

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Solution

We begin with the given function: $y=3\cos x-5\cot x$

To find $\frac{dy}{dx}$, we differentiate both sides with respect to $x$. Using the difference rule and constant multiple rule, we can differentiate each term separately while keeping their coefficients:

$\frac{dy}{dx}=3\frac{d}{dx}(\cos x)-5\frac{d}{dx}(\cot x)$

Now we substitute the standard trigonometric derivatives. Recall that:

• $\frac{d}{dx}(\cos x)=-\sin x$
• $\frac{d}{dx}(\cot x)=-{\mathrm{cosec}}^{2}x$

Applying these to our expression: $\begin{array}{rl}\frac{dy}{dx}& =3(-\sin x)-5(-{\mathrm{cosec}}^{2}x)\\ & =-3\sin x+5{\mathrm{cosec}}^{2}x\end{array}$

Thus, the derivative is $-3\sin x+5{\mathrm{cosec}}^{2}x$.

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Key Formulas or Methods Used

• Constant Multiple Rule: $\frac{d}{dx}[cf(x)]=c\frac{d}{dx}[f(x)]$
• Difference Rule: $\frac{d}{dx}[f(x)-g(x)]=\frac{d}{dx}[f(x)]-\frac{d}{dx}[g(x)]$
• Derivative of cosine: $\frac{d}{dx}(\cos x)=-\sin x$
• Derivative of cotangent: $\frac{d}{dx}(\cot x)=-{\mathrm{cosec}}^{2}x$

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Summary of Steps

1. Identify the function $y=3\cos x-5\cot x$ and apply the difference rule to separate the terms: $\frac{dy}{dx}=3\frac{d}{dx}(\cos x)-5\frac{d}{dx}(\cot x)$
2. Apply the standard derivatives: substitute $\frac{d}{dx}(\cos x)=-\sin x$ and $\frac{d}{dx}(\cot x)=-{\mathrm{cosec}}^{2}x$
3. Simplify the expression by distributing the constants: $3(-\sin x)-5(-{\mathrm{cosec}}^{2}x)=-3\sin x+5{\mathrm{cosec}}^{2}x$