Calculus Examples

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

Step 1

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

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

Step 2

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

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

Combine terms.

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Step 4.1.1

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

$x (- \sin (x)) + 0$

Step 4.1.2

Add $x (- \sin (x))$ and $0$ .

$x (- \sin (x))$

Step 4.2

Reorder the factors of $x (- \sin (x))$ .

$- x \sin (x)$