Calculus Examples

$- 2 \cot (x) + 6 x^{4} \csc (x)$

Step 1

By the Sum Rule, the derivative of $- 2 \cot (x) + 6 x^{4} \csc (x)$ with respect to $x$ is $\frac{d}{d x} [- 2 \cot (x)] + \frac{d}{d x} [6 x^{4} \csc (x)]$ .

$\frac{d}{d x} [- 2 \cot (x)] + \frac{d}{d x} [6 x^{4} \csc (x)]$

Step 2

Evaluate $\frac{d}{d x} [- 2 \cot (x)]$ .

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Step 2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 \cot (x)$ with respect to $x$ is $- 2 \frac{d}{d x} [\cot (x)]$ .

$- 2 \frac{d}{d x} [\cot (x)] + \frac{d}{d x} [6 x^{4} \csc (x)]$

Step 2.2

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

$- 2 (- \csc^{2} (x)) + \frac{d}{d x} [6 x^{4} \csc (x)]$

Step 2.3

Multiply $- 1$ by $- 2$ .

$2 \csc^{2} (x) + \frac{d}{d x} [6 x^{4} \csc (x)]$

Step 3

Evaluate $\frac{d}{d x} [6 x^{4} \csc (x)]$ .

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Step 3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{4} \csc (x)$ with respect to $x$ is $6 \frac{d}{d x} [x^{4} \csc (x)]$ .

$2 \csc^{2} (x) + 6 \frac{d}{d x} [x^{4} \csc (x)]$

Step 3.2

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

$2 \csc^{2} (x) + 6 (x^{4} \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [x^{4}])$

Step 3.3

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

$2 \csc^{2} (x) + 6 (x^{4} (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [x^{4}])$

Step 3.4

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

$2 \csc^{2} (x) + 6 (x^{4} (- \csc (x) \cot (x)) + \csc (x) (4 x^{3}))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$2 \csc^{2} (x) + 6 (x^{4} (- \csc (x) \cot (x))) + 6 (\csc (x) (4 x^{3}))$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $- 1$ by $6$ .

$2 \csc^{2} (x) - 6 (x^{4} (\csc (x) \cot (x))) + 6 (\csc (x) (4 x^{3}))$

Step 4.2.2

Multiply $4$ by $6$ .

$2 \csc^{2} (x) - 6 x^{4} (\csc (x) \cot (x)) + 24 \csc (x) x^{3}$

Step 4.3

Reorder terms.

$- 6 x^{4} \cot (x) \csc (x) + 24 x^{3} \csc (x) + 2 \csc^{2} (x)$