Calculus Examples

$4 \tan (3 x) \sec (3 x) + e^{- 2 x} + 6 x$

Step 1

By the Sum Rule, the derivative of $4 \tan (3 x) \sec (3 x) + e^{- 2 x} + 6 x$ with respect to $x$ is $\frac{d}{d x} [4 \tan (3 x) \sec (3 x)] + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$ .

$\frac{d}{d x} [4 \tan (3 x) \sec (3 x)] + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2

Evaluate $\frac{d}{d x} [4 \tan (3 x) \sec (3 x)]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \tan (3 x) \sec (3 x)$ with respect to $x$ is $4 \frac{d}{d x} [\tan (3 x) \sec (3 x)]$ .

$4 \frac{d}{d x} [\tan (3 x) \sec (3 x)] + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.2

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 $f (x) = \tan (3 x)$ and $g (x) = \sec (3 x)$ .

$4 (\tan (3 x) \frac{d}{d x} [\sec (3 x)] + \sec (3 x) \frac{d}{d x} [\tan (3 x)]) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 3 x$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$4 (\tan (3 x) (\frac{d}{d u_{1}} [\sec (u_{1})] \frac{d}{d x} [3 x]) + \sec (3 x) \frac{d}{d x} [\tan (3 x)]) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.3.2

The derivative of $\sec (u_{1})$ with respect to $u_{1}$ is $\sec (u_{1}) \tan (u_{1})$ .

$4 (\tan (3 x) (\sec (u_{1}) \tan (u_{1}) \frac{d}{d x} [3 x]) + \sec (3 x) \frac{d}{d x} [\tan (3 x)]) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $3 x$ .

$4 (\tan (3 x) (\sec (3 x) \tan (3 x) \frac{d}{d x} [3 x]) + \sec (3 x) \frac{d}{d x} [\tan (3 x)]) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$4 (\tan (3 x) (\sec (3 x) \tan (3 x) (3 \frac{d}{d x} [x])) + \sec (3 x) \frac{d}{d x} [\tan (3 x)]) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.5

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

$4 (\tan (3 x) (\sec (3 x) \tan (3 x) (3 \cdot 1)) + \sec (3 x) \frac{d}{d x} [\tan (3 x)]) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 3 x$ .

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Step 2.6.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$4 (\tan (3 x) (\sec (3 x) \tan (3 x) (3 \cdot 1)) + \sec (3 x) (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [3 x])) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.6.2

The derivative of $\tan (u_{2})$ with respect to $u_{2}$ is $\sec^{2} (u_{2})$ .

$4 (\tan (3 x) (\sec (3 x) \tan (3 x) (3 \cdot 1)) + \sec (3 x) (\sec^{2} (u_{2}) \frac{d}{d x} [3 x])) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.6.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$4 (\tan (3 x) (\sec (3 x) \tan (3 x) (3 \cdot 1)) + \sec (3 x) (\sec^{2} (3 x) \frac{d}{d x} [3 x])) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.7

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$4 (\tan (3 x) (\sec (3 x) \tan (3 x) (3 \cdot 1)) + \sec (3 x) (\sec^{2} (3 x) (3 \frac{d}{d x} [x]))) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.8

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

$4 (\tan (3 x) (\sec (3 x) \tan (3 x) (3 \cdot 1)) + \sec (3 x) (\sec^{2} (3 x) (3 \cdot 1))) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.9

Multiply $3$ by $1$ .

$4 (\tan (3 x) (\sec (3 x) \tan (3 x) \cdot 3) + \sec (3 x) (\sec^{2} (3 x) (3 \cdot 1))) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.10

Move $3$ to the left of $\sec (3 x) \tan (3 x)$ .

$4 (\tan (3 x) (3 \cdot (\sec (3 x) \tan (3 x))) + \sec (3 x) (\sec^{2} (3 x) (3 \cdot 1))) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.11

Raise $\tan (3 x)$ to the power of $1$ .

$4 (3 \sec (3 x) (\tan^{1} (3 x) \tan (3 x)) + \sec (3 x) (\sec^{2} (3 x) (3 \cdot 1))) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.12

Raise $\tan (3 x)$ to the power of $1$ .

$4 (3 \sec (3 x) (\tan^{1} (3 x) \tan^{1} (3 x)) + \sec (3 x) (\sec^{2} (3 x) (3 \cdot 1))) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.13

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

$4 (3 \sec (3 x) {\tan (3 x)}^{1 + 1} + \sec (3 x) (\sec^{2} (3 x) (3 \cdot 1))) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.14

Add $1$ and $1$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + \sec (3 x) (\sec^{2} (3 x) (3 \cdot 1))) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.15

Multiply $3$ by $1$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + \sec (3 x) (\sec^{2} (3 x) \cdot 3)) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.16

Move $3$ to the left of $\sec^{2} (3 x)$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + \sec (3 x) (3 \cdot \sec^{2} (3 x))) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.17

Raise $\sec (3 x)$ to the power of $1$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 (\sec^{1} (3 x) \sec^{2} (3 x))) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.18

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

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 {\sec (3 x)}^{1 + 2}) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 2.19

Add $1$ and $2$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) + \frac{d}{d x} [e^{- 2 x}] + \frac{d}{d x} [6 x]$

Step 3

Evaluate $\frac{d}{d x} [e^{- 2 x}]$ .

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Step 3.1

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 $g (x) = - 2 x$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{3}$ as $- 2 x$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) + \frac{d}{d u_{3}} [e^{u_{3}}] \frac{d}{d x} [- 2 x] + \frac{d}{d x} [6 x]$

Step 3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) + e^{u_{3}} \frac{d}{d x} [- 2 x] + \frac{d}{d x} [6 x]$

Step 3.1.3

Replace all occurrences of $u_{3}$ with $- 2 x$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) + e^{- 2 x} \frac{d}{d x} [- 2 x] + \frac{d}{d x} [6 x]$

Step 3.2

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

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) + e^{- 2 x} (- 2 \frac{d}{d x} [x]) + \frac{d}{d x} [6 x]$

Step 3.3

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

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) + e^{- 2 x} (- 2 \cdot 1) + \frac{d}{d x} [6 x]$

Step 3.4

Multiply $- 2$ by $1$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) + e^{- 2 x} \cdot - 2 + \frac{d}{d x} [6 x]$

Step 3.5

Move $- 2$ to the left of $e^{- 2 x}$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) - 2 e^{- 2 x} + \frac{d}{d x} [6 x]$

Step 4

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

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Step 4.1

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

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) - 2 e^{- 2 x} + 6 \frac{d}{d x} [x]$

Step 4.2

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

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) - 2 e^{- 2 x} + 6 \cdot 1$

Step 4.3

Multiply $6$ by $1$ .

$4 (3 \sec (3 x) \tan^{2} (3 x) + 3 \sec^{3} (3 x)) - 2 e^{- 2 x} + 6$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$4 (3 \sec (3 x) \tan^{2} (3 x)) + 4 (3 \sec^{3} (3 x)) - 2 e^{- 2 x} + 6$

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $3$ by $4$ .

$12 (\sec (3 x) \tan^{2} (3 x)) + 4 (3 \sec^{3} (3 x)) - 2 e^{- 2 x} + 6$

Step 5.2.2

Multiply $3$ by $4$ .

$12 \sec (3 x) \tan^{2} (3 x) + 12 \sec^{3} (3 x) - 2 e^{- 2 x} + 6$

Step 5.3

Reorder terms.

$12 \tan^{2} (3 x) \sec (3 x) + 6 - 2 e^{- 2 x} + 12 \sec^{3} (3 x)$