Calculus Examples

$x - x \sin (x)$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x \sin (x)$ with respect to $x$ is $- \frac{d}{d x} [x \sin (x)]$ .

$1 - \frac{d}{d x} [x \sin (x)]$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$1 - (x \cos (x)) - \sin (x)$

Step 3.2

Remove unnecessary parentheses.

$1 - x \cos (x) - \sin (x)$

Step 3.3

Reorder terms.

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