Question Statement

Given the parametric equations: $x=\sin 2\theta ,y=\cos 4\theta$

(a) Find $\frac{dy}{dx}$.

(b) Find $\Delta y$ and $dy$.

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Background and Explanation

This problem involves parametric differentiation, where both $x$ and $y$ are expressed in terms of a parameter $\theta$. To find $\frac{dy}{dx}$, we differentiate both equations with respect to $\theta$ and then apply the chain rule to relate these derivatives.

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Solution

Part (a): Finding $\frac{dy}{dx}$

Step 1: Differentiate $x$ with respect to $\theta$

Given: $x=\sin 2\theta$

Differentiating with respect to $\theta$ using the chain rule: $\begin{array}{rl}\frac{dx}{d\theta }& =\frac{d}{d\theta }(\sin 2\theta )\\ & =\cos 2\theta \cdot 2\\ \frac{dx}{d\theta }& =2\cos 2\theta \end{array}$

Step 2: Differentiate $y$ with respect to $\theta$

Given: $y=\cos 4\theta$

Differentiating with respect to $\theta$ using the chain rule: $\begin{array}{rl}\frac{dy}{d\theta }& =\frac{d}{d\theta }(\cos 4\theta )\\ & =-\sin 4\theta \cdot 4\\ \frac{dy}{d\theta }& =-4\sin 4\theta \end{array}$

Step 3: Apply the chain rule

By the chain rule for parametric equations: $\frac{dy}{dx}=\frac{dy}{d\theta }\times \frac{d\theta }{dx}=\frac{dy/d\theta }{dx/d\theta }$

Substituting the derivatives found above: $\begin{array}{rl}\frac{dy}{dx}& =-4\sin 4\theta \times \frac{1}{2\cos 2\theta }\\ & =\frac{-4\sin 4\theta }{2\cos 2\theta }\\ \frac{dy}{dx}& =\frac{-2\sin 4\theta }{\cos 2\theta }\end{array}$

Part (b): Finding $\Delta y$ and $dy$

(Solution steps for this part were not provided in the source data.)

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Key Formulas or Methods Used

• Chain rule for parametric equations: $\frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }$
• Derivative of sine function: $\frac{d}{d\theta }(\sin k\theta )=k\cos k\theta$
• Derivative of cosine function: $\frac{d}{d\theta }(\cos k\theta )=-k\sin k\theta$

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Summary of Steps

1. Differentiate $x=\sin 2\theta$ with respect to $\theta$ to get $\frac{dx}{d\theta }=2\cos 2\theta$.
2. Differentiate $y=\cos 4\theta$ with respect to $\theta$ to get $\frac{dy}{d\theta }=-4\sin 4\theta$.
3. Apply the chain rule: $\frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }$.
4. Substitute the expressions and simplify to obtain $\frac{dy}{dx}=\frac{-2\sin 4\theta }{\cos 2\theta }$.