Question Statement

If $\sin y=y\cos 2x$, find $\frac{dy}{dx}$.

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Background and Explanation

This problem requires implicit differentiation, where $y$ is treated as an implicit function of $x$. When differentiating terms involving $y$, we apply the chain rule by multiplying by $\frac{dy}{dx}$. The right side requires the product rule since it involves the product of two functions of $x$.

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Solution

We begin with the given equation: $\sin y=y\cos 2x$

Differentiate both sides with respect to $x$:

Step 1: Differentiate the left side using the chain rule (since $y$ is a function of $x$): $\frac{d}{dx}(\sin y)=\cos y\cdot \frac{dy}{dx}$

Step 2: Differentiate the right side $y\cos 2x$ using the product rule $\frac{d}{dx}[uv]=\frac{du}{dx}v+u\frac{dv}{dx}$: $\frac{d}{dx}(y\cos 2x)=\frac{d}{dx}(y)\cdot \cos 2x+y\cdot \frac{d}{dx}(\cos 2x)$

Step 3: Evaluate the derivatives in the product rule expression. Note that $\frac{d}{dx}(\cos 2x)=-\sin 2x\cdot 2=-2\sin 2x$ by the chain rule: $\frac{d}{dx}(y\cos 2x)=\frac{dy}{dx}\cos 2x+y(-2\sin 2x)=\frac{dy}{dx}\cos 2x-2y\sin 2x$

Step 4: Set the derivatives of both sides equal to each other: $\cos y\frac{dy}{dx}=\frac{dy}{dx}\cos 2x-2y\sin 2x$

Step 5: Collect all terms containing $\frac{dy}{dx}$ on the left side by subtracting $\frac{dy}{dx}\cos 2x$ from both sides: $\cos y\frac{dy}{dx}-\frac{dy}{dx}\cos 2x=-2y\sin 2x$

Step 6: Factor out $\frac{dy}{dx}$ from the left side: $(\cos y-\cos 2x)\frac{dy}{dx}=-2y\sin 2x$

Step 7: Solve for $\frac{dy}{dx}$ by dividing both sides by $(\cos y-\cos 2x)$: $\frac{dy}{dx}=\frac{-2y\sin 2x}{\cos y-\cos 2x}$

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Key Formulas or Methods Used

• Implicit Differentiation: $\frac{d}{dx}[f(y)]=f^{\prime }(y)\cdot \frac{dy}{dx}$ when $y$ is a function of $x$
• Chain Rule: $\frac{d}{dx}[\sin y]=\cos y\cdot \frac{dy}{dx}$ and $\frac{d}{dx}[\cos 2x]=-2\sin 2x$
• Product Rule: $\frac{d}{dx}[uv]=\frac{du}{dx}v+u\frac{dv}{dx}$

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Summary of Steps

1. Differentiate both sides of the equation $\sin y=y\cos 2x$ with respect to $x$
2. Apply the chain rule to $\sin y$ to obtain $\cos y\cdot \frac{dy}{dx}$
3. Apply the product rule to $y\cos 2x$ to obtain $\frac{dy}{dx}\cos 2x-2y\sin 2x$
4. Set the derivatives equal and collect all $\frac{dy}{dx}$ terms on the left side
5. Factor out $\frac{dy}{dx}$ from the collected terms
6. Divide by the coefficient $(\cos y-\cos 2x)$ to isolate $\frac{dy}{dx}$