Calculus Examples

$\frac{d (\frac{x^{2} \cos (x)}{\sin (x)})}{d x}$

Step 1

Convert from $\frac{x^{2} \cos (x)}{\sin (x)}$ to $x^{2} \cot (x)$ .

$\frac{d}{d x} [x^{2} \cot (x)]$

Step 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 $f (x) = x^{2}$ and $g (x) = \cot (x)$ .

$x^{2} \frac{d}{d x} [\cot (x)] + \cot (x) \frac{d}{d x} [x^{2}]$

Step 3

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) \frac{d}{d x} [x^{2}]$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x)$

Step 4.2

Reorder terms.

$- x^{2} \csc^{2} (x) + 2 x \cot (x)$