Calculus Examples

$x \cos (x)$

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

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

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

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

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

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

Combine terms.

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Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

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

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

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