Calculus Examples

$12 x \sec^{2} (6 x^{2})$

Step 1

Since $12$ is constant with respect to $x$ , the derivative of $12 x \sec^{2} (6 x^{2})$ with respect to $x$ is $12 \frac{d}{d x} [x \sec^{2} (6 x^{2})]$ .

$12 \frac{d}{d x} [x \sec^{2} (6 x^{2})]$

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 $f (x) = x$ and $g (x) = \sec^{2} (6 x^{2})$ .

$12 (x \frac{d}{d x} [\sec^{2} (6 x^{2})] + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

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 $g (x) = \sec (6 x^{2})$ .

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To apply the Chain Rule, set $u_{1}$ as $\sec (6 x^{2})$ .

$12 (x (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sec (6 x^{2})]) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

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

$12 (x (2 u_{1} \frac{d}{d x} [\sec (6 x^{2})]) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Replace all occurrences of $u_{1}$ with $\sec (6 x^{2})$ .

$12 (x (2 \sec (6 x^{2}) \frac{d}{d x} [\sec (6 x^{2})]) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 4

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 $g (x) = 6 x^{2}$ .

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To apply the Chain Rule, set $u_{2}$ as $6 x^{2}$ .

$12 (x (2 \sec (6 x^{2}) (\frac{d}{d u_{2}} [\sec (u_{2})] \frac{d}{d x} [6 x^{2}])) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

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

$12 (x (2 \sec (6 x^{2}) (\sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [6 x^{2}])) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Replace all occurrences of $u_{2}$ with $6 x^{2}$ .

$12 (x (2 \sec (6 x^{2}) (\sec (6 x^{2}) \tan (6 x^{2}) \frac{d}{d x} [6 x^{2}])) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 5

Raise $\sec (6 x^{2})$ to the power of $1$ .

$12 (x (2 (\sec^{1} (6 x^{2}) \sec (6 x^{2})) (\tan (6 x^{2}) \frac{d}{d x} [6 x^{2}])) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 6

Raise $\sec (6 x^{2})$ to the power of $1$ .

$12 (x (2 (\sec^{1} (6 x^{2}) \sec^{1} (6 x^{2})) (\tan (6 x^{2}) \frac{d}{d x} [6 x^{2}])) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 7

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

$12 (x (2 {\sec (6 x^{2})}^{1 + 1} (\tan (6 x^{2}) \frac{d}{d x} [6 x^{2}])) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 8

Add $1$ and $1$ .

$12 (x (2 \sec^{2} (6 x^{2}) (\tan (6 x^{2}) \frac{d}{d x} [6 x^{2}])) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 9

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$12 (x (2 \sec^{2} (6 x^{2}) \tan (6 x^{2}) (6 \frac{d}{d x} [x^{2}])) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 10

Multiply $6$ by $2$ .

$12 (x (12 \sec^{2} (6 x^{2}) \tan (6 x^{2}) \frac{d}{d x} [x^{2}]) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 11

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

$12 (x (12 \sec^{2} (6 x^{2}) \tan (6 x^{2}) (2 x)) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 12

Multiply $2$ by $12$ .

$12 (x (24 \sec^{2} (6 x^{2}) \tan (6 x^{2}) x) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 13

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

$12 (x^{1} x (24 \sec^{2} (6 x^{2}) \tan (6 x^{2})) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 14

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

$12 (x^{1} x^{1} (24 \sec^{2} (6 x^{2}) \tan (6 x^{2})) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 15

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

$12 (x^{1 + 1} (24 \sec^{2} (6 x^{2}) \tan (6 x^{2})) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 16

Add $1$ and $1$ .

$12 (x^{2} (24 \sec^{2} (6 x^{2}) \tan (6 x^{2})) + \sec^{2} (6 x^{2}) \frac{d}{d x} [x])$

Step 17

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

$12 (x^{2} (24 \sec^{2} (6 x^{2}) \tan (6 x^{2})) + \sec^{2} (6 x^{2}) \cdot 1)$

Step 18

Multiply $\sec^{2} (6 x^{2})$ by $1$ .

$12 (x^{2} (24 \sec^{2} (6 x^{2}) \tan (6 x^{2})) + \sec^{2} (6 x^{2}))$

Step 19

Simplify.

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Apply the distributive property.

$12 (x^{2} (24 \sec^{2} (6 x^{2}) \tan (6 x^{2}))) + 12 \sec^{2} (6 x^{2})$

Multiply $24$ by $12$ .

$288 x^{2} (\sec^{2} (6 x^{2}) \tan (6 x^{2})) + 12 \sec^{2} (6 x^{2})$

Reorder terms.

$288 x^{2} \sec^{2} (6 x^{2}) \tan (6 x^{2}) + 12 \sec^{2} (6 x^{2})$