Solve for the one with 1,2,3 and 4

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ScanSolve AI Tutor · Accepted Answer

Here's the solution for the question with options 1, 2, 3, and 4 (Question 63).

63. If $y = \sin(x^2)$ , find $\frac{dy}{dx}$ .

Identify the functions for the chain rule.
Let $u = x^2$ . Then $y = \sin(u)$ .
Find the derivatives of $y$ with respect to $u$ and $u$ with respect to $x$ .
$\frac{dy}{du} = \frac{d}{du}(\sin u) = \cos u$ $\frac{du}{dx} = \frac{d}{dx}(x^2) = 2x$
Apply the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ .
$\frac{dy}{dx} = (\cos u) \cdot (2x)$
Substitute $u = x^2$ back into the expression.
$\frac{dy}{dx} = 2x \cos(x^2)$
The correct option is 1. $2x \cos(x^2)$ .

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