Calculus Examples

$f (x) = - \frac{3}{\sin^{2} (7 x)}$

Step 1

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{\sin^{2} (7 x)}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{\sin^{2} (7 x)}]$ .

$- 3 \frac{d}{d x} [\frac{1}{\sin^{2} (7 x)}]$

Step 2

Convert from $\frac{1}{\sin^{2} (7 x)}$ to $\csc^{2} (7 x)$ .

$- 3 \frac{d}{d x} [\csc^{2} (7 x)]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \csc (7 x)$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $\csc (7 x)$ .

$- 3 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\csc (7 x)])$

Step 3.2

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

$- 3 (2 u_{1} \frac{d}{d x} [\csc (7 x)])$

Step 3.3

Replace all occurrences of $u_{1}$ with $\csc (7 x)$ .

$- 3 (2 \csc (7 x) \frac{d}{d x} [\csc (7 x)])$

Step 4

Multiply $2$ by $- 3$ .

$- 6 (\csc (7 x) \frac{d}{d x} [\csc (7 x)])$

Step 5

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 5.1

To apply the Chain Rule, set $u_{2}$ as $7 x$ .

$- 6 \csc (7 x) (\frac{d}{d u_{2}} [\csc (u_{2})] \frac{d}{d x} [7 x])$

Step 5.2

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

$- 6 \csc (7 x) (- \csc (u_{2}) \cot (u_{2}) \frac{d}{d x} [7 x])$

Step 5.3

Replace all occurrences of $u_{2}$ with $7 x$ .

$- 6 \csc (7 x) (- \csc (7 x) \cot (7 x) \frac{d}{d x} [7 x])$

Step 6

Multiply $- 1$ by $- 6$ .

$6 \csc (7 x) (\csc (7 x) \cot (7 x) \frac{d}{d x} [7 x])$

Step 7

Raise $\csc (7 x)$ to the power of $1$ .

$6 (\csc^{1} (7 x) \csc (7 x)) (\cot (7 x) \frac{d}{d x} [7 x])$

Step 8

Raise $\csc (7 x)$ to the power of $1$ .

$6 (\csc^{1} (7 x) \csc^{1} (7 x)) (\cot (7 x) \frac{d}{d x} [7 x])$

Step 9

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

$6 {\csc (7 x)}^{1 + 1} (\cot (7 x) \frac{d}{d x} [7 x])$

Step 10

Add $1$ and $1$ .

$6 \csc^{2} (7 x) (\cot (7 x) \frac{d}{d x} [7 x])$

Step 11

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

$6 \csc^{2} (7 x) \cot (7 x) (7 \frac{d}{d x} [x])$

Step 12

Multiply $7$ by $6$ .

$42 \csc^{2} (7 x) \cot (7 x) \frac{d}{d x} [x]$

Step 13

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

$42 \csc^{2} (7 x) \cot (7 x) \cdot 1$

Step 14

Multiply $42$ by $1$ .

$42 \csc^{2} (7 x) \cot (7 x)$