Calculus Examples

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

Step 1

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

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

Step 2

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

Tap for more steps...

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

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $3$ by $1$ .

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

Step 2.5

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

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

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

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

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

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

Step 3.3.3

Replace all occurrences of $u_{2}$ with $\sec (3 x)$ .

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

Step 3.4

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 $g (x) = 3 x$ .

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

The derivative of $\sec (u_{3})$ with respect to $u_{3}$ is $\sec (u_{3}) \tan (u_{3})$ .

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

Step 3.4.3

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

$3 \sec^{2} (3 x) + 3 (x (3 \sec^{2} (3 x) (\sec (3 x) \tan (3 x) (3 \cdot 1))) + \sec^{3} (3 x) \cdot 1)$

Step 3.8

Multiply $3$ by $1$ .

$3 \sec^{2} (3 x) + 3 (x (3 \sec^{2} (3 x) (\sec (3 x) \tan (3 x) \cdot 3)) + \sec^{3} (3 x) \cdot 1)$

Step 3.9

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

$3 \sec^{2} (3 x) + 3 (x (3 \sec^{2} (3 x) (3 \cdot (\sec (3 x) \tan (3 x)))) + \sec^{3} (3 x) \cdot 1)$

Step 3.10

Multiply $3$ by $3$ .

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

Step 3.11

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

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

Step 3.12

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

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

Step 3.13

Add $1$ and $2$ .

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

Step 3.14

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $9$ by $3$ .

$3 \sec^{2} (3 x) + 27 x (\sec^{3} (3 x) \tan (3 x)) + 3 \sec^{3} (3 x)$

Step 4.3

Reorder terms.

$27 x \sec^{3} (3 x) \tan (3 x) + 3 \sec^{2} (3 x) + 3 \sec^{3} (3 x)$