Calculus Examples

$11 x^{4} \cot (x)$

Step 1

Since $11$ is constant with respect to $x$ , the derivative of $11 x^{4} \cot (x)$ with respect to $x$ is $11 \frac{d}{d x} [x^{4} \cot (x)]$ .

$11 \frac{d}{d x} [x^{4} \cot (x)]$

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 $f (x) = x^{4}$ and $g (x) = \cot (x)$ .

$11 (x^{4} \frac{d}{d x} [\cot (x)] + \cot (x) \frac{d}{d x} [x^{4}])$

Step 3

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

$11 (x^{4} (- \csc^{2} (x)) + \cot (x) \frac{d}{d x} [x^{4}])$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$11 (x^{4} (- \csc^{2} (x)) + \cot (x) (4 x^{3}))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$11 (x^{4} (- \csc^{2} (x))) + 11 (\cot (x) (4 x^{3}))$

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $- 1$ by $11$ .

$- 11 (x^{4} (\csc^{2} (x))) + 11 (\cot (x) (4 x^{3}))$

Step 5.2.2

Multiply $4$ by $11$ .

$- 11 x^{4} \csc^{2} (x) + 44 \cot (x) x^{3}$

Step 5.3

Reorder terms.

$- 11 x^{4} \csc^{2} (x) + 44 x^{3} \cot (x)$