Calculus Examples

$y = \sec^{3} (2 x)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \sec (2 x)$ .

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Step 1.1

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

$\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\sec (2 x)]$

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 $n = 3$ .

$3 {u_{1}}^{2} \frac{d}{d x} [\sec (2 x)]$

Step 1.3

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Raise $\sec (2 x)$ to the power of $1$ .

$3 (\sec^{1} (2 x) \sec^{2} (2 x)) (\tan (2 x) \frac{d}{d x} [2 x])$

Step 4

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

$3 {\sec (2 x)}^{1 + 2} (\tan (2 x) \frac{d}{d x} [2 x])$

Step 5

Add $1$ and $2$ .

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

Step 6

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

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

Step 7

Multiply $2$ by $3$ .

$6 \sec^{3} (2 x) \tan (2 x) \frac{d}{d x} [x]$

Step 8

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

$6 \sec^{3} (2 x) \tan (2 x) \cdot 1$

Step 9

Multiply $6$ by $1$ .

$6 \sec^{3} (2 x) \tan (2 x)$