Given the function y=x-x sin x, find (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, (d/dx)(x sin x) = (d/dx)(x) * sin x + x * (d/dx)(sin x).Differentiate x and sin x: Differentiate x and sin x separately. The derivative of x with respect to x is 1. The derivative of sin x with respect to x is cos x. So, (d/dx)(x) = 1 and (d/dx)(sin x) = cos x.Substitute derivatives into formula: Substitute the derivatives into the product rule formula. From Step 1, we have (d/dx)(x sin x) = 1 * sin x + x * cos x. This simplifies to (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, (dy/dx) = (d/dx)(x) - (d/dx)(x sin x).Substitute derivatives into equation: Substitute the derivatives found in Steps 2 and 3 into the equation from Step 4. We have (dy/dx) = 1 - (sin x + x cos x). This simplifies to (dy/dx) = 1 - sin x - x cos x.

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Algebra 2

Sin, cos, and tan of special angles

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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 $(\frac{dy}{dx}) = 1 - \sin x - x \cos x$ .