Question Statement

Q8. Find the derivative of the function $f(x)=\tan \left(\cos \frac{x}{2}\right)$ with respect to $x$.

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Background and Explanation

This problem involves differentiating a composite function using the chain rule multiple times. You will need to apply the derivatives of tangent and cosine functions sequentially, working from the outermost function inward.

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Solution

Given the function: $f(x)=\tan \left(\cos \frac{x}{2}\right)$

We differentiate with respect to $x$ using the chain rule. The function is a composition where the outer function is $\tan (u)$ and the inner function is $u=\cos \left(\frac{x}{2}\right)$.

Applying the chain rule: $\begin{array}{rl}{f}^{\prime }(x)& ={\sec }^2\left(\cos \frac{x}{2}\right)\cdot \frac{d}{dx}\left(\cos \frac{x}{2}\right)\\ & ={\sec }^2\left(\cos \frac{x}{2}\right)\cdot \left(-\sin \frac{x}{2}\right)\cdot \frac{d}{dx}\left(\frac{x}{2}\right)\\ & =-{\sec }^2\left(\cos \frac{x}{2}\right)\left(\sin \frac{x}{2}\right)\cdot \frac{1}{2}\\ & =\frac{-1}{2}{\sec }^2\left(\cos \frac{x}{2}\right)\cdot \sin \left(\frac{x}{2}\right)\end{array}$

Step-by-step reasoning:

1. First chain rule application: The derivative of $\tan (u)$ is ${\sec }^2(u)$, so we get ${\sec }^2\left(\cos \frac{x}{2}\right)$ multiplied by the derivative of the inner function $\cos \frac{x}{2}$.
2. Second chain rule application: To differentiate $\cos \frac{x}{2}$, we apply the chain rule again. The derivative of $\cos (v)$ is $-\sin (v)$, giving us $-\sin \frac{x}{2}$ multiplied by the derivative of $\frac{x}{2}$.
3. Constant multiple rule: The derivative of $\frac{x}{2}$ is $\frac{1}{2}$.
4. Simplification: Combine all factors, moving the constant $\frac{1}{2}$ to the front and keeping the negative sign.

Use implicit differentiation to find $\frac{dy}{dx}$ :

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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 tangent: $\frac{d}{du}[\tan (u)]={\sec }^2(u)$
• Derivative of cosine: $\frac{d}{dv}[\cos (v)]=-\sin (v)$
• Constant Multiple Rule: $\frac{d}{dx}[cx]=c$ where $c$ is a constant

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Summary of Steps

1. Identify the outermost function as $\tan (u)$ and apply its derivative ${\sec }^2(u)$
2. Multiply by the derivative of the inner function $\cos \left(\frac{x}{2}\right)$ using the chain rule again
3. Differentiate $\frac{x}{2}$ to get $\frac{1}{2}$ and multiply by $-\sin \left(\frac{x}{2}\right)$
4. Combine all terms and simplify the expression to $-\frac{1}{2}{\sec }^2\left(\cos \frac{x}{2}\right)\sin \left(\frac{x}{2}\right)$