Calculus Examples

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

Step 1

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

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

Step 2

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

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Add $1$ and $1$ .

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

Step 3

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

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 4

Combine terms.

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Step 4.1

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

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

Step 4.2

Add $2 \sec^{2} (x) \tan (x)$ and $2 \sec^{2} (x) \tan (x)$ .

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