Calculus Examples

$\tan (x)$

Step 1

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

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

Step 2

Find the second derivative.

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

Tap for more steps...

Step 2.1.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

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

Step 2.1.2

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

$2 u \frac{d}{d x} [\sec (x)]$

Step 2.1.3

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 2.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$2 \sec (x) (\sec (x) \tan (x))$

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Add $1$ and $1$ .

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

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

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

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

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

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

Step 3.3

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

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

Step 3.4

Multiply $\sec^{2} (x)$ by $\sec^{2} (x)$ by adding the exponents.

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

Add $2$ and $2$ .

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

Step 3.5

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

Tap for more steps...

Step 3.5.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

$2 (\sec^{4} (x) + \tan (x) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\sec (x)]))$

Step 3.5.2

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

$2 (\sec^{4} (x) + \tan (x) (2 u \frac{d}{d x} [\sec (x)]))$

Step 3.5.3

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 3.6

Move $2$ to the left of $\tan (x)$ .

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

Step 3.7

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$2 (\sec^{4} (x) + 2 \tan (x) \sec (x) (\sec (x) \tan (x)))$

Step 3.8

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

$2 (\sec^{4} (x) + 2 (\tan^{1} (x) \tan (x)) \sec (x) \sec (x))$

Step 3.9

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

$2 (\sec^{4} (x) + 2 (\tan^{1} (x) \tan^{1} (x)) \sec (x) \sec (x))$

Step 3.10

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

$2 (\sec^{4} (x) + 2 {\tan (x)}^{1 + 1} \sec (x) \sec (x))$

Step 3.11

Add $1$ and $1$ .

$2 (\sec^{4} (x) + 2 \tan^{2} (x) \sec (x) \sec (x))$

Step 3.12

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

$2 (\sec^{4} (x) + 2 \tan^{2} (x) (\sec^{1} (x) \sec (x)))$

Step 3.13

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

$2 (\sec^{4} (x) + 2 \tan^{2} (x) (\sec^{1} (x) \sec^{1} (x)))$

Step 3.14

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

$2 (\sec^{4} (x) + 2 \tan^{2} (x) {\sec (x)}^{1 + 1})$

Step 3.15

Add $1$ and $1$ .

$2 (\sec^{4} (x) + 2 \tan^{2} (x) \sec^{2} (x))$

Step 3.16

Simplify.

Tap for more steps...

Step 3.16.1

Apply the distributive property.

$2 \sec^{4} (x) + 2 (2 \tan^{2} (x) \sec^{2} (x))$

Step 3.16.2

Multiply $2$ by $2$ .

$2 \sec^{4} (x) + 4 \tan^{2} (x) \sec^{2} (x)$

Step 3.16.3

Reorder terms.

$f''' (x) = 4 \sec^{2} (x) \tan^{2} (x) + 2 \sec^{4} (x)$