Calculus Examples

$f (x) = x^{2} \sec^{2} (3 x)$

Step 1

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

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

Step 2

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 $g (x) = \sec (3 x)$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{1}$ with $\sec (3 x)$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

Multiply $3$ by $2$ .

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

Step 10

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

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

Step 11

Multiply $6$ by $1$ .

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

Step 12

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

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

Step 13

Reorder terms.

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