Question Statement

Find the differential $dy$ for the function: $y=\sin x$

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Background and Explanation

Differentials provide a linear approximation of how a function changes with small changes in its input. For a function $y=f(x)$, the differential is defined as $dy=f^{\prime }(x)dx$, representing the principal linear part of the increment $\Delta y$.

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Solution

To find the differential of the function $y=\sin x$, we can look at the relationship between the increment $\Delta y$ and the differential $dy$.

Step 1: Determine the increment $\Delta y$

Given the function: $y=\sin x$

We start by expressing the function after an increment $\Delta x$ is added to the input: $y+\Delta y=\sin (x+\Delta x)$

To isolate the change in $y$ ($\Delta y$), we subtract the original function from both sides: $\begin{array}{rl}\Delta y& =\sin (x+\Delta x)-y\\ & =\sin (x+\Delta x)-\sin x\end{array}$

Step 2: Apply trigonometric identities

Using the trigonometric identity for the difference of sines, $\sin A-\sin B=2\cos \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)$, we can rewrite the expression for $\Delta y$: $\begin{array}{rl}\Delta y& =2\cos \left(\frac{(x+\Delta x)+x}{2}\right)\sin \left(\frac{(x+\Delta x)-x}{2}\right)\\ & =2\cos \left(\frac{2x+\Delta x}{2}\right)\sin \left(\frac{\Delta x}{2}\right)\end{array}$

Step 3: Find the differential $dy$

By definition, the differential $dy$ is the product of the derivative of the function and the differential of the independent variable $dx$.

For the function $y=\sin x$, we find the derivative: $\frac{dy}{dx}=\cos x$

Applying the formula for the differential $dy=f^{\prime }(x)dx$: $\begin{array}{rl}dy& =d(\sin x)\\ & =\cos xdx\end{array}$

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Key Formulas or Methods Used

• Definition of Differential: $dy=f^{\prime }(x)dx$
• Increment Method: $\Delta y=f(x+\Delta x)-f(x)$
• Trigonometric Identity: $\sin A-\sin B=2\cos \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)$
• Derivative of Sine: $\frac{d}{dx}(\sin x)=\cos x$

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Summary of Steps

1. Identify the function: Start with $y=\sin x$.
2. Analyze the increment: Use the sine difference identity to show that $\Delta y=2\cos \left(\frac{2x+\Delta x}{2}\right)\sin \left(\frac{\Delta x}{2}\right)$.
3. Calculate the derivative: Determine that $f^{\prime }(x)=\cos x$.
4. State the differential: Multiply the derivative by $dx$ to get $dy=\cos xdx$.