Calculus Examples

$f (θ) = 3 \tan^{2} (θ)$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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

$3 \frac{d}{d θ} [\tan^{2} (θ)]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

To apply the Chain Rule, set $u$ as $\tan (θ)$ .

$3 (\frac{d}{d u} [u^{2}] \frac{d}{d θ} [\tan (θ)])$

Step 1.2.2

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

$3 (2 u \frac{d}{d θ} [\tan (θ)])$

Step 1.2.3

Replace all occurrences of $u$ with $\tan (θ)$ .

$3 (2 \tan (θ) \frac{d}{d θ} [\tan (θ)])$

Step 1.3

Multiply $2$ by $3$ .

$6 (\tan (θ) \frac{d}{d θ} [\tan (θ)])$

Step 1.4

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

$6 \tan (θ) \sec^{2} (θ)$

Step 1.5

Reorder the factors of $6 \tan (θ) \sec^{2} (θ)$ .

$f' (θ) = 6 \sec^{2} (θ) \tan (θ)$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

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

$6 \frac{d}{d θ} [\sec^{2} (θ) \tan (θ)]$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d θ} [f (θ) g (θ)]$ is $f (θ) \frac{d}{d θ} [g (θ)] + g (θ) \frac{d}{d θ} [f (θ)]$ where $f (θ) = \sec^{2} (θ)$ and $g (θ) = \tan (θ)$ .

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

Step 2.3

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

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

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

Add $2$ and $2$ .

$6 (\sec^{4} (θ) + \tan (θ) \frac{d}{d θ} [\sec^{2} (θ)])$

Step 2.5

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

Tap for more steps...

Step 2.5.1

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

$6 (\sec^{4} (θ) + \tan (θ) (\frac{d}{d u} [u^{2}] \frac{d}{d θ} [\sec (θ)]))$

Step 2.5.2

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

$6 (\sec^{4} (θ) + \tan (θ) (2 u \frac{d}{d θ} [\sec (θ)]))$

Step 2.5.3

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

$6 (\sec^{4} (θ) + \tan (θ) (2 \sec (θ) \frac{d}{d θ} [\sec (θ)]))$

Step 2.6

Move $2$ to the left of $\tan (θ)$ .

$6 (\sec^{4} (θ) + 2 \cdot \tan (θ) \sec (θ) \frac{d}{d θ} [\sec (θ)])$

Step 2.7

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

$6 (\sec^{4} (θ) + 2 \tan (θ) \sec (θ) (\sec (θ) \tan (θ)))$

Step 2.8

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

$6 (\sec^{4} (θ) + 2 (\tan^{1} (θ) \tan (θ)) \sec (θ) \sec (θ))$

Step 2.9

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

$6 (\sec^{4} (θ) + 2 (\tan^{1} (θ) \tan^{1} (θ)) \sec (θ) \sec (θ))$

Step 2.10

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

$6 (\sec^{4} (θ) + 2 {\tan (θ)}^{1 + 1} \sec (θ) \sec (θ))$

Step 2.11

Add $1$ and $1$ .

$6 (\sec^{4} (θ) + 2 \tan^{2} (θ) \sec (θ) \sec (θ))$

Step 2.12

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

$6 (\sec^{4} (θ) + 2 \tan^{2} (θ) (\sec^{1} (θ) \sec (θ)))$

Step 2.13

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

$6 (\sec^{4} (θ) + 2 \tan^{2} (θ) (\sec^{1} (θ) \sec^{1} (θ)))$

Step 2.14

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

$6 (\sec^{4} (θ) + 2 \tan^{2} (θ) {\sec (θ)}^{1 + 1})$

Step 2.15

Add $1$ and $1$ .

$6 (\sec^{4} (θ) + 2 \tan^{2} (θ) \sec^{2} (θ))$

Step 2.16

Simplify.

Tap for more steps...

Step 2.16.1

Apply the distributive property.

$6 \sec^{4} (θ) + 6 (2 \tan^{2} (θ) \sec^{2} (θ))$

Step 2.16.2

Multiply $2$ by $6$ .

$6 \sec^{4} (θ) + 12 \tan^{2} (θ) \sec^{2} (θ)$

Step 2.16.3

Reorder terms.

$f'' (θ) = 12 \sec^{2} (θ) \tan^{2} (θ) + 6 \sec^{4} (θ)$

Step 3

The second derivative of $f (θ)$ with respect to $θ$ is $12 \sec^{2} (θ) \tan^{2} (θ) + 6 \sec^{4} (θ)$ .

$12 \sec^{2} (θ) \tan^{2} (θ) + 6 \sec^{4} (θ)$