Calculus Examples

$\sec^{2} (x - 1)$

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 $g (x) = \sec (x - 1)$ .

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Step 1.1

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

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

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 $n = 2$ .

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

Step 1.3

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

$2 \sec (x - 1) \frac{d}{d x} [\sec (x - 1)]$

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 $g (x) = x - 1$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $x - 1$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{2}$ with $x - 1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 8

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

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

Step 9

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 \sec^{2} (x - 1) \tan (x - 1) (1 + 0)$

Step 10

Simplify the expression.

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Step 10.1

Add $1$ and $0$ .

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

Step 10.2

Multiply $2$ by $1$ .

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