Calculus Examples

$e^{6 x} - \csc (x) \cot (x)$

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [e^{6 x}]$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 6 x$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $6 x$ .

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

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 2.1.3

Replace all occurrences of $u$ with $6 x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $6$ by $1$ .

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

Step 2.5

Move $6$ to the left of $e^{6 x}$ .

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

Step 3

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

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \csc (x) \cot (x)$ with respect to $x$ is $- \frac{d}{d x} [\csc (x) \cot (x)]$ .

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

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

$6 e^{6 x} - (- (\csc^{1} (x) \csc^{2} (x)) + \cot (x) (- \csc (x) \cot (x)))$

Step 3.6

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

$6 e^{6 x} - (- {\csc (x)}^{1 + 2} + \cot (x) (- \csc (x) \cot (x)))$

Step 3.7

Add $1$ and $2$ .

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

Step 3.8

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

$6 e^{6 x} - (- \csc^{3} (x) - \csc (x) (\cot^{1} (x) \cot (x)))$

Step 3.9

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

$6 e^{6 x} - (- \csc^{3} (x) - \csc (x) (\cot^{1} (x) \cot^{1} (x)))$

Step 3.10

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

$6 e^{6 x} - (- \csc^{3} (x) - \csc (x) {\cot (x)}^{1 + 1})$

Step 3.11

Add $1$ and $1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$6 e^{6 x} + 1 \csc^{3} (x) - (- \csc (x) \cot^{2} (x))$

Step 4.2.2

Multiply $\csc^{3} (x)$ by $1$ .

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

Step 4.2.3

Multiply $- 1$ by $- 1$ .

$6 e^{6 x} + \csc^{3} (x) + 1 (\csc (x) \cot^{2} (x))$

Step 4.2.4

Multiply $\csc (x)$ by $1$ .

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

Step 4.3

Reorder terms.

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