Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $x \sin (x) + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [x \sin (x)] + \frac{d}{d x} [\cos (x)]$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x \cos (x) + \sin (x) \cdot 1 + \frac{d}{d x} [\cos (x)]$

Step 2.4

Multiply $\sin (x)$ by $1$ .

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

Step 3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 4

Combine terms.

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Step 4.1

Subtract $\sin (x)$ from $\sin (x)$ .

$x \cos (x) + 0$

Step 4.2

Add $x \cos (x)$ and $0$ .

$x \cos (x)$