Calculus Examples

$y = \sqrt{\sec (4 x)}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\sec (4 x)}$ as ${\sec (4 x)}^{\frac{1}{2}}$ .

$y = {\sec (4 x)}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({\sec (4 x)}^{\frac{1}{2}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = \sec (4 x)$ .

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Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as $\sec (4 x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [\sec (4 x)]$

Step 4.1.2

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [\sec (4 x)]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $\sec (4 x)$ .

$\frac{1}{2} {\sec (4 x)}^{\frac{1}{2} - 1} \frac{d}{d x} [\sec (4 x)]$

Step 4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {\sec (4 x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\sec (4 x)]$

Step 4.3

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} {\sec (4 x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\sec (4 x)]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{1}{2} {\sec (4 x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [\sec (4 x)]$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {\sec (4 x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [\sec (4 x)]$

Step 4.5.2

Subtract $2$ from $1$ .

$\frac{1}{2} {\sec (4 x)}^{\frac{- 1}{2}} \frac{d}{d x} [\sec (4 x)]$

Step 4.6

Combine fractions.

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Step 4.6.1

Move the negative in front of the fraction.

$\frac{1}{2} {\sec (4 x)}^{- \frac{1}{2}} \frac{d}{d x} [\sec (4 x)]$

Step 4.6.2

Combine $\frac{1}{2}$ and ${\sec (4 x)}^{- \frac{1}{2}}$ .

$\frac{{\sec (4 x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [\sec (4 x)]$

Step 4.6.3

Move ${\sec (4 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {\sec (4 x)}^{\frac{1}{2}}} \frac{d}{d x} [\sec (4 x)]$

Step 4.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 4 x$ .

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Step 4.7.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

$\frac{1}{2 {\sec (4 x)}^{\frac{1}{2}}} (\frac{d}{d u_{2}} [\sec (u_{2})] \frac{d}{d x} [4 x])$

Step 4.7.2

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

$\frac{1}{2 {\sec (4 x)}^{\frac{1}{2}}} (\sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [4 x])$

Step 4.7.3

Replace all occurrences of $u_{2}$ with $4 x$ .

$\frac{1}{2 {\sec (4 x)}^{\frac{1}{2}}} (\sec (4 x) \tan (4 x) \frac{d}{d x} [4 x])$

Step 4.8

Combine fractions.

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Step 4.8.1

Combine $\sec (4 x)$ and $\frac{1}{2 {\sec (4 x)}^{\frac{1}{2}}}$ .

$\frac{\sec (4 x)}{2 {\sec (4 x)}^{\frac{1}{2}}} (\tan (4 x) \frac{d}{d x} [4 x])$

Step 4.8.2

Combine $\tan (4 x)$ and $\frac{\sec (4 x)}{2 {\sec (4 x)}^{\frac{1}{2}}}$ .

$\frac{\tan (4 x) \sec (4 x)}{2 {\sec (4 x)}^{\frac{1}{2}}} \frac{d}{d x} [4 x]$

Step 4.8.3

Move ${\sec (4 x)}^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{\tan (4 x) \sec (4 x) {\sec (4 x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 x]$

Step 4.9

Multiply $\sec (4 x)$ by ${\sec (4 x)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.9.1

Move ${\sec (4 x)}^{- \frac{1}{2}}$ .

$\frac{\tan (4 x) ({\sec (4 x)}^{- \frac{1}{2}} \sec (4 x))}{2} \frac{d}{d x} [4 x]$

Step 4.9.2

Multiply ${\sec (4 x)}^{- \frac{1}{2}}$ by $\sec (4 x)$ .

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Step 4.9.2.1

Raise $\sec (4 x)$ to the power of $1$ .

$\frac{\tan (4 x) ({\sec (4 x)}^{- \frac{1}{2}} \sec^{1} (4 x))}{2} \frac{d}{d x} [4 x]$

Step 4.9.2.2

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

$\frac{\tan (4 x) {\sec (4 x)}^{- \frac{1}{2} + 1}}{2} \frac{d}{d x} [4 x]$

Step 4.9.3

Write $1$ as a fraction with a common denominator.

$\frac{\tan (4 x) {\sec (4 x)}^{- \frac{1}{2} + \frac{2}{2}}}{2} \frac{d}{d x} [4 x]$

Step 4.9.4

Combine the numerators over the common denominator.

$\frac{\tan (4 x) {\sec (4 x)}^{\frac{- 1 + 2}{2}}}{2} \frac{d}{d x} [4 x]$

Step 4.9.5

Add $- 1$ and $2$ .

$\frac{\tan (4 x) {\sec (4 x)}^{\frac{1}{2}}}{2} \frac{d}{d x} [4 x]$

Step 4.10

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{\tan (4 x) {\sec (4 x)}^{\frac{1}{2}}}{2} (4 \frac{d}{d x} [x])$

Step 4.11

Simplify terms.

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Step 4.11.1

Combine $4$ and $\frac{\tan (4 x) {\sec (4 x)}^{\frac{1}{2}}}{2}$ .

$\frac{4 (\tan (4 x) {\sec (4 x)}^{\frac{1}{2}})}{2} \frac{d}{d x} [x]$

Step 4.11.2

Factor $2$ out of $4 \tan (4 x) {\sec (4 x)}^{\frac{1}{2}}$ .

$\frac{2 (2 \tan (4 x) {\sec (4 x)}^{\frac{1}{2}})}{2} \frac{d}{d x} [x]$

Step 4.12

Cancel the common factors.

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Step 4.12.1

Factor $2$ out of $2$ .

$\frac{2 (2 \tan (4 x) {\sec (4 x)}^{\frac{1}{2}})}{2 (1)} \frac{d}{d x} [x]$

Step 4.12.2

Cancel the common factor.

$\frac{2 (2 \tan (4 x) {\sec (4 x)}^{\frac{1}{2}})}{2 \cdot 1} \frac{d}{d x} [x]$

Step 4.12.3

Rewrite the expression.

$\frac{2 \tan (4 x) {\sec (4 x)}^{\frac{1}{2}}}{1} \frac{d}{d x} [x]$

Step 4.12.4

Divide $2 \tan (4 x) {\sec (4 x)}^{\frac{1}{2}}$ by $1$ .

$2 \tan (4 x) {\sec (4 x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 4.13

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

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

Step 4.14

Simplify the expression.

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Step 4.14.1

Multiply $2$ by $1$ .

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

Step 4.14.2

Reorder the factors of $2 \tan (4 x) {\sec (4 x)}^{\frac{1}{2}}$ .

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

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = 2 {\sec (4 x)}^{\frac{1}{2}} \tan (4 x)$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 2 {\sec (4 x)}^{\frac{1}{2}} \tan (4 x)$