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Given the function $y=x-x\sin x$ , find $\frac{dy}{dx}$ in any form.

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Sin, cos, and tan of special angles

Full solution

Q. Given the function

y=x-x\sin x

, find

\frac{dy}{dx}

in any form.

Apply product rule: Apply the product rule to the term $x \sin x$ . The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. So, $\frac{d}{dx}(x \sin x) = \frac{d}{dx}(x) \cdot \sin x + x \cdot \frac{d}{dx}(\sin x)$ .
Differentiate $x$ and $\sin x$ : Differentiate $x$ and $\sin x$ separately. $\newline$ The derivative of $x$ with respect to $x$ is $1$ . $\newline$ The derivative of $\sin x$ with respect to $x$ is $\cos x$ . $\newline$ So, $\sin x$ $0$ and $\sin x$ $1$ .
Substitute derivatives into formula: Substitute the derivatives into the product rule formula. $\newline$ From Step $1$ , we have $(\frac{d}{dx})(x \sin x) = 1 \cdot \sin x + x \cdot \cos x$ . $\newline$ This simplifies to $(\frac{d}{dx})(x \sin x) = \sin x + x \cos x$ .
Differentiate entire function: Differentiate the entire function $y = x - x \sin x$ . The derivative of $y$ with respect to $x$ is the derivative of $x$ minus the derivative of $x \sin x$ . So, $\frac{dy}{dx} = \frac{d}{dx}(x) - \frac{d}{dx}(x \sin x)$ .
Substitute derivatives into equation: Substitute the derivatives found in Steps $2$ and $3$ into the equation from Step $4$ . $\newline$ We have $(\frac{dy}{dx}) = 1 - (\sin x + x \cos x)$ . $\newline$ This simplifies to $(\frac{dy}{dx}) = 1 - \sin x - x \cos x$ .

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