Calculus Examples

$(9 x \sec (3 x^{2} - 7)) (\tan (3 x))$

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $9 x \sec (3 x^{2} - 7) \tan (3 x)$ with respect to $x$ is $9 \frac{d}{d x} [x \sec (3 x^{2} - 7) \tan (3 x)]$ .

$9 \frac{d}{d x} [x \sec (3 x^{2} - 7) \tan (3 x)]$

Step 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 $f (x) = x \sec (3 x^{2} - 7)$ and $g (x) = \tan (3 x)$ .

$9 (x \sec (3 x^{2} - 7) \frac{d}{d x} [\tan (3 x)] + \tan (3 x) \frac{d}{d x} [x \sec (3 x^{2} - 7)])$

Step 3

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

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Step 3.1

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

$9 (x \sec (3 x^{2} - 7) (\frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d x} [3 x]) + \tan (3 x) \frac{d}{d x} [x \sec (3 x^{2} - 7)])$

Step 3.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

$9 (x \sec (3 x^{2} - 7) (\sec^{2} (u_{1}) \frac{d}{d x} [3 x]) + \tan (3 x) \frac{d}{d x} [x \sec (3 x^{2} - 7)])$

Step 3.3

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

$9 (x \sec (3 x^{2} - 7) (\sec^{2} (3 x) \frac{d}{d x} [3 x]) + \tan (3 x) \frac{d}{d x} [x \sec (3 x^{2} - 7)])$

Step 4

Differentiate.

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Step 4.1

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

$9 (x \sec (3 x^{2} - 7) \sec^{2} (3 x) (3 \frac{d}{d x} [x]) + \tan (3 x) \frac{d}{d x} [x \sec (3 x^{2} - 7)])$

Step 4.2

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

$9 (x \sec (3 x^{2} - 7) \sec^{2} (3 x) (3 \cdot 1) + \tan (3 x) \frac{d}{d x} [x \sec (3 x^{2} - 7)])$

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $3$ by $1$ .

$9 (x \sec (3 x^{2} - 7) \sec^{2} (3 x) \cdot 3 + \tan (3 x) \frac{d}{d x} [x \sec (3 x^{2} - 7)])$

Step 4.3.2

Move $3$ to the left of $x \sec (3 x^{2} - 7) \sec^{2} (3 x)$ .

$9 (3 \cdot (x \sec (3 x^{2} - 7) \sec^{2} (3 x)) + \tan (3 x) \frac{d}{d x} [x \sec (3 x^{2} - 7)])$

Step 5

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 $f (x) = x$ and $g (x) = \sec (3 x^{2} - 7)$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x \frac{d}{d x} [\sec (3 x^{2} - 7)] + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 3 x^{2} - 7$ .

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Step 6.1

To apply the Chain Rule, set $u_{2}$ as $3 x^{2} - 7$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x (\frac{d}{d u_{2}} [\sec (u_{2})] \frac{d}{d x} [3 x^{2} - 7]) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 6.2

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

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x (\sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [3 x^{2} - 7]) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 6.3

Replace all occurrences of $u_{2}$ with $3 x^{2} - 7$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) \frac{d}{d x} [3 x^{2} - 7]) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 7

Differentiate.

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Step 7.1

By the Sum Rule, the derivative of $3 x^{2} - 7$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 7]$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 7])) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 7.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7])) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 7.3

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

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) (3 (2 x) + \frac{d}{d x} [- 7])) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 7.4

Multiply $2$ by $3$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) (6 x + \frac{d}{d x} [- 7])) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 7.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) (6 x + 0)) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 7.6

Add $6 x$ and $0$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) (6 x)) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 8

Raise $x$ to the power of $1$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x^{1} x (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) \cdot (6)) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 9

Raise $x$ to the power of $1$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x^{1} x^{1} (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) \cdot (6)) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 10

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

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x^{1 + 1} (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) \cdot (6)) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 11

Simplify the expression.

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Step 11.1

Add $1$ and $1$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x^{2} (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7) \cdot (6)) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 11.2

Move $6$ to the left of $\sec (3 x^{2} - 7) \tan (3 x^{2} - 7)$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x^{2} (6 \cdot (\sec (3 x^{2} - 7) \tan (3 x^{2} - 7))) + \sec (3 x^{2} - 7) \frac{d}{d x} [x]))$

Step 12

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

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x^{2} (6 \sec (3 x^{2} - 7) \tan (3 x^{2} - 7)) + \sec (3 x^{2} - 7) \cdot 1))$

Step 13

Multiply $\sec (3 x^{2} - 7)$ by $1$ .

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x^{2} (6 \sec (3 x^{2} - 7) \tan (3 x^{2} - 7)) + \sec (3 x^{2} - 7)))$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + \tan (3 x) (x^{2} (6 \sec (3 x^{2} - 7) \tan (3 x^{2} - 7))) + \tan (3 x) \sec (3 x^{2} - 7))$

Step 14.2

Apply the distributive property.

$9 (3 x \sec (3 x^{2} - 7) \sec^{2} (3 x)) + 9 (\tan (3 x) (x^{2} (6 \sec (3 x^{2} - 7) \tan (3 x^{2} - 7)))) + 9 (\tan (3 x) \sec (3 x^{2} - 7))$

Step 14.3

Combine terms.

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Step 14.3.1

Multiply $3$ by $9$ .

$27 (x \sec (3 x^{2} - 7) \sec^{2} (3 x)) + 9 (\tan (3 x) (x^{2} (6 \sec (3 x^{2} - 7) \tan (3 x^{2} - 7)))) + 9 (\tan (3 x) \sec (3 x^{2} - 7))$

Step 14.3.2

Move $6$ to the left of $\tan (3 x) x^{2}$ .

$27 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + 9 (6 \cdot (\tan (3 x) x^{2}) \sec (3 x^{2} - 7) \tan (3 x^{2} - 7)) + 9 (\tan (3 x) \sec (3 x^{2} - 7))$

Step 14.3.3

Multiply $6$ by $9$ .

$27 x \sec (3 x^{2} - 7) \sec^{2} (3 x) + 54 \tan (3 x) x^{2} \sec (3 x^{2} - 7) \tan (3 x^{2} - 7) + 9 \tan (3 x) \sec (3 x^{2} - 7)$

Step 14.4

Reorder terms.

$27 x \sec^{2} (3 x) \sec (3 x^{2} - 7) + 54 x^{2} \sec (3 x^{2} - 7) \tan (3 x) \tan (3 x^{2} - 7) + 9 \sec (3 x^{2} - 7) \tan (3 x)$