Calculus Examples

$\frac{5}{\sin (7 x)}$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{\sin (7 x)}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{\sin (7 x)}]$ .

$5 \frac{d}{d x} [\frac{1}{\sin (7 x)}]$

Step 2

Convert from $\frac{1}{\sin (7 x)}$ to $\csc (7 x)$ .

$5 \frac{d}{d x} [\csc (7 x)]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \csc (x)$ and $g (x) = 7 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $7 x$ .

$5 (\frac{d}{d u} [\csc (u)] \frac{d}{d x} [7 x])$

Step 3.2

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

$5 (- \csc (u) \cot (u) \frac{d}{d x} [7 x])$

Step 3.3

Replace all occurrences of $u$ with $7 x$ .

$5 (- \csc (7 x) \cot (7 x) \frac{d}{d x} [7 x])$

Step 4

Differentiate.

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Step 4.1

Multiply $- 1$ by $5$ .

$- 5 (\csc (7 x) \cot (7 x) \frac{d}{d x} [7 x])$

Step 4.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$- 5 \csc (7 x) \cot (7 x) (7 \frac{d}{d x} [x])$

Step 4.3

Multiply $7$ by $- 5$ .

$- 35 \csc (7 x) \cot (7 x) \frac{d}{d x} [x]$

Step 4.4

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

$- 35 \csc (7 x) \cot (7 x) \cdot 1$

Step 4.5

Simplify the expression.

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Step 4.5.1

Multiply $- 35$ by $1$ .

$- 35 \csc (7 x) \cot (7 x)$

Step 4.5.2

Reorder the factors of $- 35 \csc (7 x) \cot (7 x)$ .

$- 35 \cot (7 x) \csc (7 x)$