Calculus Examples

$\frac{d^{2}}{d x^{2}} (5 \csc (x))$

Step 1

This derivative could not be completed using the product rule. Mathway will use another method.

Step 2

Find the first derivative.

Tap for more steps...

Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \csc (x)$ with respect to $x$ is $5 \frac{d}{d x} [\csc (x)]$ .

$5 \frac{d}{d x} [\csc (x)]$

Step 2.2

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

$5 (- \csc (x) \cot (x))$

Step 2.3

Simplify the expression.

Tap for more steps...

Step 2.3.1

Multiply $- 1$ by $5$ .

$- 5 (\csc (x) \cot (x))$

Step 2.3.2

Reorder the factors of $- 5 \csc (x) \cot (x)$ .

$f' (x) = - 5 \cot (x) \csc (x)$

Step 3

Find the second derivative.

Tap for more steps...

Step 3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 \cot (x) \csc (x)$ with respect to $x$ is $- 5 \frac{d}{d x} [\cot (x) \csc (x)]$ .

$- 5 \frac{d}{d x} [\cot (x) \csc (x)]$

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 $f (x) = \cot (x)$ and $g (x) = \csc (x)$ .

$- 5 (\cot (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 3.3

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

$- 5 (\cot (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 3.4

Raise $\cot (x)$ to the power of $1$ .

$- 5 (- \csc (x) (\cot^{1} (x) \cot (x)) + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 3.5

Raise $\cot (x)$ to the power of $1$ .

$- 5 (- \csc (x) (\cot^{1} (x) \cot^{1} (x)) + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 5 (- \csc (x) {\cot (x)}^{1 + 1} + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 3.7

Add $1$ and $1$ .

$- 5 (- \csc (x) \cot^{2} (x) + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 3.8

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

$- 5 (- \csc (x) \cot^{2} (x) + \csc (x) (- \csc^{2} (x)))$

Step 3.9

Raise $\csc (x)$ to the power of $1$ .

$- 5 (- \csc (x) \cot^{2} (x) - (\csc^{1} (x) \csc^{2} (x)))$

Step 3.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 5 (- \csc (x) \cot^{2} (x) - {\csc (x)}^{1 + 2})$

Step 3.11

Add $1$ and $2$ .

$- 5 (- \csc (x) \cot^{2} (x) - \csc^{3} (x))$

Step 3.12

Simplify.

Tap for more steps...

Step 3.12.1

Apply the distributive property.

$- 5 (- \csc (x) \cot^{2} (x)) - 5 (- \csc^{3} (x))$

Step 3.12.2

Combine terms.

Tap for more steps...

Step 3.12.2.1

Multiply $- 1$ by $- 5$ .

$5 (\csc (x) \cot^{2} (x)) - 5 (- \csc^{3} (x))$

Step 3.12.2.2

Multiply $- 1$ by $- 5$ .

$5 \csc (x) \cot^{2} (x) + 5 \csc^{3} (x)$

Step 3.12.3

Reorder terms.

$f'' (x) = 5 \cot^{2} (x) \csc (x) + 5 \csc^{3} (x)$