Calculus Examples

$y = \sqrt{6 + \sec (π x^{2})}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 + \sec (π x^{2})}$ as ${(6 + \sec (π x^{2}))}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(6 + \sec (π x^{2}))}^{\frac{1}{2}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 6 + \sec (π x^{2})$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $6 + \sec (π x^{2})$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 2.2

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 2.3

Replace all occurrences of $u_{1}$ with $6 + \sec (π x^{2})$ .

$\frac{1}{2} {(6 + \sec (π x^{2}))}^{\frac{1}{2} - 1} \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 3

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

$\frac{1}{2} {(6 + \sec (π x^{2}))}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 4

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

$\frac{1}{2} {(6 + \sec (π x^{2}))}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{2} {(6 + \sec (π x^{2}))}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(6 + \sec (π x^{2}))}^{\frac{1 - 2}{2}} \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(6 + \sec (π x^{2}))}^{\frac{- 1}{2}} \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 7

Differentiate.

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Step 7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(6 + \sec (π x^{2}))}^{- \frac{1}{2}} \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 7.2

Combine fractions.

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Step 7.2.1

Combine $\frac{1}{2}$ and ${(6 + \sec (π x^{2}))}^{- \frac{1}{2}}$ .

$\frac{{(6 + \sec (π x^{2}))}^{- \frac{1}{2}}}{2} \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 7.2.2

Move ${(6 + \sec (π x^{2}))}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} \frac{d}{d x} [6 + \sec (π x^{2})]$

Step 7.3

By the Sum Rule, the derivative of $6 + \sec (π x^{2})$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [\sec (π x^{2})]$ .

$\frac{1}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} (\frac{d}{d x} [6] + \frac{d}{d x} [\sec (π x^{2})])$

Step 7.4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{1}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} (0 + \frac{d}{d x} [\sec (π x^{2})])$

Step 7.5

Add $0$ and $\frac{d}{d x} [\sec (π x^{2})]$ .

$\frac{1}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} \frac{d}{d x} [\sec (π x^{2})]$

Step 8

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 $g (x) = π x^{2}$ .

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Step 8.1

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

$\frac{1}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} (\frac{d}{d u_{2}} [\sec (u_{2})] \frac{d}{d x} [π x^{2}])$

Step 8.2

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

$\frac{1}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} (\sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [π x^{2}])$

Step 8.3

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

$\frac{1}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} (\sec (π x^{2}) \tan (π x^{2}) \frac{d}{d x} [π x^{2}])$

Step 9

Differentiate.

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Step 9.1

Combine $\sec (π x^{2})$ and $\frac{1}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}}$ .

$\frac{\sec (π x^{2})}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} (\tan (π x^{2}) \frac{d}{d x} [π x^{2}])$

Step 9.2

Combine $\tan (π x^{2})$ and $\frac{\sec (π x^{2})}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}}$ .

$\frac{\tan (π x^{2}) \sec (π x^{2})}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} \frac{d}{d x} [π x^{2}]$

Step 9.3

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

$\frac{\tan (π x^{2}) \sec (π x^{2})}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} (π \frac{d}{d x} [x^{2}])$

Step 9.4

Combine $π$ and $\frac{\tan (π x^{2}) \sec (π x^{2})}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}}$ .

$\frac{π (\tan (π x^{2}) \sec (π x^{2}))}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}]$

Step 9.5

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

$\frac{π \tan (π x^{2}) \sec (π x^{2})}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} (2 x)$

Step 9.6

Simplify terms.

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Step 9.6.1

Combine $2$ and $\frac{π \tan (π x^{2}) \sec (π x^{2})}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}}$ .

$\frac{2 (π \tan (π x^{2}) \sec (π x^{2}))}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}} x$

Step 9.6.2

Combine $\frac{2 (π \tan (π x^{2}) \sec (π x^{2}))}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}}$ and $x$ .

$\frac{2 (π \tan (π x^{2}) \sec (π x^{2})) x}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}}$

Step 9.6.3

Cancel the common factor.

$\frac{2 π \tan (π x^{2}) \sec (π x^{2}) x}{2 {(6 + \sec (π x^{2}))}^{\frac{1}{2}}}$

Step 9.6.4

Rewrite the expression.

$\frac{π \tan (π x^{2}) \sec (π x^{2}) x}{{(6 + \sec (π x^{2}))}^{\frac{1}{2}}}$

Step 10

Simplify.

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Step 10.1

Reorder factors in $π \tan (π x^{2}) \sec (π x^{2}) x$ .

$\frac{π x \tan (π x^{2}) \sec (π x^{2})}{{(6 + \sec (π x^{2}))}^{\frac{1}{2}}}$

Step 10.2

Reorder terms.

$\frac{π x \sec (π x^{2}) \tan (π x^{2})}{{(6 + \sec (π x^{2}))}^{\frac{1}{2}}}$