Calculus Examples

$\sec^{2} (e^{2 x})$

Step 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 $g (x) = \sec (e^{2 x})$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sec (e^{2 x})$ .

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

Step 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 $n = 2$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\sec (e^{2 x})$ .

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $e^{2 x}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{2}$ with $e^{2 x}$ .

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

Step 3

Raise $\sec (e^{2 x})$ to the power of $1$ .

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

Step 4

Raise $\sec (e^{2 x})$ to the power of $1$ .

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

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

Step 7.3

Replace all occurrences of $u_{3}$ with $2 x$ .

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

Step 8

Differentiate.

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Step 8.1

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

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

Step 8.2

Multiply $2$ by $2$ .

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

Step 8.3

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

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

Step 8.4

Simplify the expression.

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Step 8.4.1

Multiply $e^{2 x}$ by $1$ .

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

Step 8.4.2

Reorder the factors of $4 \sec^{2} (e^{2 x}) \tan (e^{2 x}) e^{2 x}$ .

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