Question Statement

Find the derivative of $y=4{\cot }^{-1}\left(\frac{x}{2}\right)$ with respect to $x$.

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Background and Explanation

This problem requires applying the chain rule for composite functions along with the standard derivative formula for inverse trigonometric functions. Specifically, you'll need to differentiate the outer inverse cotangent function while accounting for the inner linear function $\frac{x}{2}$.

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Solution

Given the function: $y=4{\cot }^{-1}\left(\frac{x}{2}\right)$

To find $\frac{dy}{dx}$, we apply the chain rule. Recall that if $y=4{\cot }^{-1}(u)$ where $u=\frac{x}{2}$, then:

$\frac{dy}{dx}=4\cdot \frac{d}{du}({\cot }^{-1}u)\cdot \frac{du}{dx}$

Step 1: Differentiate the outer function (inverse cotangent) and inner function separately.

• The derivative of ${\cot }^{-1}(u)$ with respect to $u$ is $\frac{-1}{1+u^2}$
• The derivative of $u=\frac{x}{2}$ with respect to $x$ is $\frac{1}{2}$

Step 2: Apply the chain rule formula. $\frac{dy}{dx}=4\cdot \frac{-1}{1+{\left(\frac{x}{2}\right)}^2}\cdot \frac{d}{dx}\left(\frac{x}{2}\right)$

Step 3: Substitute the derivative of the inner function. $=4\cdot \frac{-1}{1+\frac{x^2}{4}}\cdot \frac{1}{2}$

Step 4: Simplify the constant coefficients ($4\times \frac{1}{2}=2$). $=2\cdot \frac{-1}{\frac{4+x^2}{4}}$

Step 5: Simplify the complex fraction by multiplying by the reciprocal of the denominator. $=-2\cdot \frac{4}{4+x^2}$

Step 6: Final multiplication to obtain the simplified result. $\frac{dy}{dx}=\frac{-8}{4+x^2}$

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Key Formulas or Methods Used

• Chain Rule: $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$
• Derivative of Inverse Cotangent: $\frac{d}{du}({\cot }^{-1}u)=\frac{-1}{1+u^2}$
• Power/Linear Rule: $\frac{d}{dx}\left(\frac{x}{2}\right)=\frac{1}{2}$

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Summary of Steps

1. Identify components: Recognize the outer function $4{\cot }^{-1}(u)$ and inner function $u=\frac{x}{2}$
2. Apply chain rule: Multiply the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function
3. Substitute values: Replace $\frac{d}{dx}\left(\frac{x}{2}\right)$ with $\frac{1}{2}$ and simplify the denominator $1+{\left(\frac{x}{2}\right)}^2$ to $\frac{4+x^2}{4}$
4. Simplify coefficients: Combine $4\times \frac{1}{2}=2$ to reduce the expression
5. Clear the fraction: Multiply by the reciprocal to resolve $\frac{1}{\frac{4+x^2}{4}}=\frac{4}{4+x^2}$
6. Final calculation: Multiply $-2\times 4$ to obtain the final answer $\frac{-8}{4+x^2}$