Calculus Examples

$y = \arctan (\sqrt{7 x^{2} - 1})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x^{2} - 1}$ as ${(7 x^{2} - 1)}^{\frac{1}{2}}$ .

$y = \arctan ({(7 x^{2} - 1)}^{\frac{1}{2}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arctan ({(7 x^{2} - 1)}^{\frac{1}{2}}))$

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 $f (x) = \arctan (x)$ and $g (x) = {(7 x^{2} - 1)}^{\frac{1}{2}}$ .

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Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as ${(7 x^{2} - 1)}^{\frac{1}{2}}$ .

$\frac{d}{d u_{1}} [\arctan (u_{1})] \frac{d}{d x} [{(7 x^{2} - 1)}^{\frac{1}{2}}]$

Step 4.1.2

The derivative of $\arctan (u_{1})$ with respect to $u_{1}$ is $\frac{1}{1 + {u_{1}}^{2}}$ .

$\frac{1}{1 + {u_{1}}^{2}} \frac{d}{d x} [{(7 x^{2} - 1)}^{\frac{1}{2}}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with ${(7 x^{2} - 1)}^{\frac{1}{2}}$ .

$\frac{1}{1 + {({(7 x^{2} - 1)}^{\frac{1}{2}})}^{2}} \frac{d}{d x} [{(7 x^{2} - 1)}^{\frac{1}{2}}]$

Step 4.2

Multiply the exponents in ${({(7 x^{2} - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{1 + {(7 x^{2} - 1)}^{\frac{1}{2} \cdot 2}} \frac{d}{d x} [{(7 x^{2} - 1)}^{\frac{1}{2}}]$

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{1}{1 + {(7 x^{2} - 1)}^{\frac{1}{2} \cdot 2}} \frac{d}{d x} [{(7 x^{2} - 1)}^{\frac{1}{2}}]$

Step 4.2.2.2

Rewrite the expression.

$\frac{1}{1 + {(7 x^{2} - 1)}^{1}} \frac{d}{d x} [{(7 x^{2} - 1)}^{\frac{1}{2}}]$

Step 4.3

Simplify.

$\frac{1}{1 + 7 x^{2} - 1} \frac{d}{d x} [{(7 x^{2} - 1)}^{\frac{1}{2}}]$

Step 4.4

Simplify by subtracting numbers.

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Step 4.4.1

Subtract $1$ from $1$ .

$\frac{1}{7 x^{2} + 0} \frac{d}{d x} [{(7 x^{2} - 1)}^{\frac{1}{2}}]$

Step 4.4.2

Add $7 x^{2}$ and $0$ .

$\frac{1}{7 x^{2}} \frac{d}{d x} [{(7 x^{2} - 1)}^{\frac{1}{2}}]$

Step 4.5

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 $g (x) = 7 x^{2} - 1$ .

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Step 4.5.1

To apply the Chain Rule, set $u_{2}$ as $7 x^{2} - 1$ .

$\frac{1}{7 x^{2}} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [7 x^{2} - 1])$

Step 4.5.2

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

$\frac{1}{7 x^{2}} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [7 x^{2} - 1])$

Step 4.5.3

Replace all occurrences of $u_{2}$ with $7 x^{2} - 1$ .

$\frac{1}{7 x^{2}} (\frac{1}{2} {(7 x^{2} - 1)}^{\frac{1}{2} - 1} \frac{d}{d x} [7 x^{2} - 1])$

Step 4.6

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

$\frac{1}{7 x^{2}} (\frac{1}{2} {(7 x^{2} - 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [7 x^{2} - 1])$

Step 4.7

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

$\frac{1}{7 x^{2}} (\frac{1}{2} {(7 x^{2} - 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [7 x^{2} - 1])$

Step 4.8

Combine the numerators over the common denominator.

$\frac{1}{7 x^{2}} (\frac{1}{2} {(7 x^{2} - 1)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [7 x^{2} - 1])$

Step 4.9

Simplify the numerator.

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Step 4.9.1

Multiply $- 1$ by $2$ .

$\frac{1}{7 x^{2}} (\frac{1}{2} {(7 x^{2} - 1)}^{\frac{1 - 2}{2}} \frac{d}{d x} [7 x^{2} - 1])$

Step 4.9.2

Subtract $2$ from $1$ .

$\frac{1}{7 x^{2}} (\frac{1}{2} {(7 x^{2} - 1)}^{\frac{- 1}{2}} \frac{d}{d x} [7 x^{2} - 1])$

Step 4.10

Combine fractions.

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Step 4.10.1

Move the negative in front of the fraction.

$\frac{1}{7 x^{2}} (\frac{1}{2} {(7 x^{2} - 1)}^{- \frac{1}{2}} \frac{d}{d x} [7 x^{2} - 1])$

Step 4.10.2

Combine $\frac{1}{2}$ and ${(7 x^{2} - 1)}^{- \frac{1}{2}}$ .

$\frac{1}{7 x^{2}} (\frac{{(7 x^{2} - 1)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [7 x^{2} - 1])$

Step 4.10.3

Move ${(7 x^{2} - 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{7 x^{2}} (\frac{1}{2 {(7 x^{2} - 1)}^{\frac{1}{2}}} \frac{d}{d x} [7 x^{2} - 1])$

Step 4.10.4

Multiply $\frac{1}{2 {(7 x^{2} - 1)}^{\frac{1}{2}}}$ by $\frac{1}{7 x^{2}}$ .

$\frac{1}{2 {(7 x^{2} - 1)}^{\frac{1}{2}} (7 x^{2})} \frac{d}{d x} [7 x^{2} - 1]$

Step 4.10.5

Multiply $7$ by $2$ .

$\frac{1}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}} \frac{d}{d x} [7 x^{2} - 1]$

Step 4.11

By the Sum Rule, the derivative of $7 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [- 1]$ .

$\frac{1}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}} (\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [- 1])$

Step 4.12

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}} (7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])$

Step 4.13

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

$\frac{1}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}} (7 (2 x) + \frac{d}{d x} [- 1])$

Step 4.14

Multiply $2$ by $7$ .

$\frac{1}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}} (14 x + \frac{d}{d x} [- 1])$

Step 4.15

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

$\frac{1}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}} (14 x + 0)$

Step 4.16

Simplify terms.

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Step 4.16.1

Add $14 x$ and $0$ .

$\frac{1}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}} (14 x)$

Step 4.16.2

Combine $14$ and $\frac{1}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}}$ .

$\frac{14}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}} x$

Step 4.16.3

Combine $\frac{14}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}}$ and $x$ .

$\frac{14 x}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}}$

Step 4.16.4

Cancel the common factor.

$\frac{14 x}{14 {(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}}$

Step 4.16.5

Rewrite the expression.

$\frac{x}{{(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}}$

Step 4.17

Raise $x$ to the power of $1$ .

$\frac{x^{1}}{{(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}}$

Step 4.18

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1}{{(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}}$

Step 4.19

Cancel the common factors.

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Step 4.19.1

Factor $x$ out of ${(7 x^{2} - 1)}^{\frac{1}{2}} x^{2}$ .

$\frac{x \cdot 1}{x ({(7 x^{2} - 1)}^{\frac{1}{2}} x)}$

Step 4.19.2

Cancel the common factor.

$\frac{x \cdot 1}{x ({(7 x^{2} - 1)}^{\frac{1}{2}} x)}$

Step 4.19.3

Rewrite the expression.

$\frac{1}{{(7 x^{2} - 1)}^{\frac{1}{2}} x}$

Step 4.20

Reorder terms.

$\frac{1}{x {(7 x^{2} - 1)}^{\frac{1}{2}}}$

Step 5

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

$y' = \frac{1}{x {(7 x^{2} - 1)}^{\frac{1}{2}}}$

Step 6

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

$\frac{d y}{d x} = \frac{1}{x {(7 x^{2} - 1)}^{\frac{1}{2}}}$