Calculus Examples

y = x sin (x)

$y = x \sin (x)$

Step 1

Differentiate both sides of the equation.

\frac{d}{d x} (y) = \frac{d}{d x} (x sin (x))

$\frac{d}{d x} (y) = \frac{d}{d x} (x \sin (x))$

Step 2

The derivative of

y

$y$ with respect to

x

$x$ is

$y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that

$\frac{d}{d x} [f (x) g (x)]$ is

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

$f (x) = x$ and

$g (x) = \sin (x)$ .

$x \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x]$

Step 3.2

The derivative of

$\sin (x)$ with respect to

$x$ is

$\cos (x)$ .

$x \cos (x) + \sin (x) \frac{d}{d x} [x]$

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$x \cos (x) + \sin (x) \cdot 1$

Step 3.3.2

Multiply

$\sin (x)$ by

$1$ .

$x \cos (x) + \sin (x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = x \cos (x) + \sin (x)$

Step 5

Replace

$y'$ with

$\frac{d y}{d x}$ .

$\frac{d y}{d x} = x \cos (x) + \sin (x)$