Calculus Examples

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

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

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

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

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Step 2.1

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

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

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 $n = 2$ .

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

Move $\sec^{2} (x)$ .

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

Step 5.2

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

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

Step 5.3

Add $2$ and $2$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Add $1$ and $2$ .

$2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec (x) \sec (x)$

Step 12

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

$2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) (\sec^{1} (x) \sec (x))$

Step 13

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

$2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) (\sec^{1} (x) \sec^{1} (x))$

Step 14

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

$2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) {\sec (x)}^{1 + 1}$

Step 15

Add $1$ and $1$ .

$2 \sec^{4} (x) \tan (x) + 2 \tan^{3} (x) \sec^{2} (x)$