Calculus Examples

$\sqrt{1 - 49 x^{2}} \arccos (7 x)$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 49 x^{2}}$ as ${(1 - 49 x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}} \arccos (7 x)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(1 - 49 x^{2})}^{\frac{1}{2}}$ and $g (x) = \arccos (7 x)$ .

${(1 - 49 x^{2})}^{\frac{1}{2}} \frac{d}{d x} [\arccos (7 x)] + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arccos (x)$ and $g (x) = 7 x$ .

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Step 3.1

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

${(1 - 49 x^{2})}^{\frac{1}{2}} (\frac{d}{d u_{1}} [\arccos (u_{1})] \frac{d}{d x} [7 x]) + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 3.2

The derivative of $\arccos (u_{1})$ with respect to $u_{1}$ is $- \frac{1}{\sqrt{1 - {u_{1}}^{2}}}$ .

${(1 - 49 x^{2})}^{\frac{1}{2}} (- \frac{1}{\sqrt{1 - {u_{1}}^{2}}} \frac{d}{d x} [7 x]) + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 3.3

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

${(1 - 49 x^{2})}^{\frac{1}{2}} (- \frac{1}{\sqrt{1 - {(7 x)}^{2}}} \frac{d}{d x} [7 x]) + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4

Differentiate.

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Step 4.1

Factor $7$ out of $7 x$ .

${(1 - 49 x^{2})}^{\frac{1}{2}} (- \frac{1}{\sqrt{1 - {(7 (x))}^{2}}} \frac{d}{d x} [7 x]) + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4.2

Combine fractions.

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Step 4.2.1

Simplify the expression.

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Step 4.2.1.1

Apply the product rule to $7 (x)$ .

${(1 - 49 x^{2})}^{\frac{1}{2}} (- \frac{1}{\sqrt{1 - (7^{2} x^{2})}} \frac{d}{d x} [7 x]) + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4.2.1.2

Raise $7$ to the power of $2$ .

${(1 - 49 x^{2})}^{\frac{1}{2}} (- \frac{1}{\sqrt{1 - (49 x^{2})}} \frac{d}{d x} [7 x]) + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4.2.1.3

Multiply $49$ by $- 1$ .

${(1 - 49 x^{2})}^{\frac{1}{2}} (- \frac{1}{\sqrt{1 - 49 x^{2}}} \frac{d}{d x} [7 x]) + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4.2.2

Combine ${(1 - 49 x^{2})}^{\frac{1}{2}}$ and $\frac{1}{\sqrt{1 - 49 x^{2}}}$ .

$- \frac{{(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} \frac{d}{d x} [7 x] + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4.3

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

$- \frac{{(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} (7 \frac{d}{d x} [x]) + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4.4

Combine fractions.

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Step 4.4.1

Multiply $7$ by $- 1$ .

$- 7 \frac{{(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} \frac{d}{d x} [x] + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4.4.2

Combine $- 7$ and $\frac{{(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}}$ .

$\frac{- 7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} \frac{d}{d x} [x] + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4.4.3

Move the negative in front of the fraction.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} \frac{d}{d x} [x] + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4.5

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

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} \cdot 1 + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 4.6

Multiply $- 1$ by $1$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) \frac{d}{d x} [{(1 - 49 x^{2})}^{\frac{1}{2}}]$

Step 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 $g (x) = 1 - 49 x^{2}$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $1 - 49 x^{2}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [1 - 49 x^{2}])$

Step 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 $n = \frac{1}{2}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [1 - 49 x^{2}])$

Step 5.3

Replace all occurrences of $u_{2}$ with $1 - 49 x^{2}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{1}{2} {(1 - 49 x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [1 - 49 x^{2}])$

Step 6

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

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{1}{2} {(1 - 49 x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [1 - 49 x^{2}])$

Step 7

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

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{1}{2} {(1 - 49 x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [1 - 49 x^{2}])$

Step 8

Combine the numerators over the common denominator.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{1}{2} {(1 - 49 x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [1 - 49 x^{2}])$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{1}{2} {(1 - 49 x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [1 - 49 x^{2}])$

Step 9.2

Subtract $2$ from $1$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{1}{2} {(1 - 49 x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [1 - 49 x^{2}])$

Step 10

Combine fractions.

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Step 10.1

Move the negative in front of the fraction.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{1}{2} {(1 - 49 x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [1 - 49 x^{2}])$

Step 10.2

Combine $\frac{1}{2}$ and ${(1 - 49 x^{2})}^{- \frac{1}{2}}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{{(1 - 49 x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [1 - 49 x^{2}])$

Step 10.3

Move ${(1 - 49 x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \arccos (7 x) (\frac{1}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [1 - 49 x^{2}])$

Step 10.4

Combine $\frac{1}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}}$ and $\arccos (7 x)$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{\arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [1 - 49 x^{2}]$

Step 11

By the Sum Rule, the derivative of $1 - 49 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 49 x^{2}]$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{\arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [1] + \frac{d}{d x} [- 49 x^{2}])$

Step 12

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

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{\arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- 49 x^{2}])$

Step 13

Add $0$ and $\frac{d}{d x} [- 49 x^{2}]$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{\arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- 49 x^{2}]$

Step 14

Since $- 49$ is constant with respect to $x$ , the derivative of $- 49 x^{2}$ with respect to $x$ is $- 49 \frac{d}{d x} [x^{2}]$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{\arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} (- 49 \frac{d}{d x} [x^{2}])$

Step 15

Combine fractions.

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Step 15.1

Combine $- 49$ and $\frac{\arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{- 49 \arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}]$

Step 15.2

Move the negative in front of the fraction.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} - \frac{49 \arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}]$

Step 16

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

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} - \frac{49 \arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} (2 x)$

Step 17

Simplify terms.

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Step 17.1

Multiply $2$ by $- 1$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} - 2 \frac{49 \arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} x$

Step 17.2

Combine $- 2$ and $\frac{49 \arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{- 2 (49 \arccos (7 x))}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} x$

Step 17.3

Multiply $49$ by $- 2$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{- 98 \arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}} x$

Step 17.4

Combine $\frac{- 98 \arccos (7 x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}}$ and $x$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{- 98 \arccos (7 x) x}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}}$

Step 17.5

Factor $2$ out of $- 98 \arccos (7 x) x$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{2 (- 49 \arccos (7 x) x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}}$

Step 18

Cancel the common factors.

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Step 18.1

Factor $2$ out of $2 {(1 - 49 x^{2})}^{\frac{1}{2}}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{2 (- 49 \arccos (7 x) x)}{2 ({(1 - 49 x^{2})}^{\frac{1}{2}})}$

Step 18.2

Cancel the common factor.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{2 (- 49 \arccos (7 x) x)}{2 {(1 - 49 x^{2})}^{\frac{1}{2}}}$

Step 18.3

Rewrite the expression.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} + \frac{- 49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}}$

Step 19

Move the negative in front of the fraction.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} - \frac{49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}}$

Step 20

To write $- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{\frac{1}{2}}}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}} \cdot \frac{{(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{\frac{1}{2}}} - \frac{49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}}$

Step 21

To write $- \frac{49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{1 - 49 x^{2}}}{\sqrt{1 - 49 x^{2}}}$ .

Step 22

Write each expression with a common denominator of $\sqrt{1 - 49 x^{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 22.1

Multiply $\frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}}}$ by $\frac{{(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{\frac{1}{2}}}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{\sqrt{1 - 49 x^{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}} - \frac{49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}} \cdot \frac{\sqrt{1 - 49 x^{2}}}{\sqrt{1 - 49 x^{2}}}$

Step 22.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 49 x^{2}}$ as ${(1 - 49 x^{2})}^{\frac{1}{2}}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}} - \frac{49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}} \cdot \frac{\sqrt{1 - 49 x^{2}}}{\sqrt{1 - 49 x^{2}}}$

Step 22.3

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

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{\frac{1}{2} + \frac{1}{2}}} - \frac{49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}} \cdot \frac{\sqrt{1 - 49 x^{2}}}{\sqrt{1 - 49 x^{2}}}$

Step 22.4

Combine the numerators over the common denominator.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{\frac{1 + 1}{2}}} - \frac{49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}} \cdot \frac{\sqrt{1 - 49 x^{2}}}{\sqrt{1 - 49 x^{2}}}$

Step 22.5

Add $1$ and $1$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{\frac{2}{2}}} - \frac{49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}} \cdot \frac{\sqrt{1 - 49 x^{2}}}{\sqrt{1 - 49 x^{2}}}$

Step 22.6

Cancel the common factor of $2$ .

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Step 22.6.1

Cancel the common factor.

Step 22.6.2

Rewrite the expression.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{1}} - \frac{49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}} \cdot \frac{\sqrt{1 - 49 x^{2}}}{\sqrt{1 - 49 x^{2}}}$

Step 22.7

Multiply $\frac{49 \arccos (7 x) x}{{(1 - 49 x^{2})}^{\frac{1}{2}}}$ by $\frac{\sqrt{1 - 49 x^{2}}}{\sqrt{1 - 49 x^{2}}}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{1}} - \frac{49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{\frac{1}{2}} \sqrt{1 - 49 x^{2}}}$

Step 22.8

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - 49 x^{2}}$ as ${(1 - 49 x^{2})}^{\frac{1}{2}}$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{1}} - \frac{49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}$

Step 22.9

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

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{1}} - \frac{49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{\frac{1}{2} + \frac{1}{2}}}$

Step 22.10

Combine the numerators over the common denominator.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{1}} - \frac{49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{\frac{1 + 1}{2}}}$

Step 22.11

Add $1$ and $1$ .

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{1}} - \frac{49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{\frac{2}{2}}}$

Step 22.12

Cancel the common factor of $2$ .

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Step 22.12.1

Cancel the common factor.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{1}} - \frac{49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{\frac{2}{2}}}$

Step 22.12.2

Rewrite the expression.

$- \frac{7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}}{{(1 - 49 x^{2})}^{1}} - \frac{49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{1}}$

Step 23

Combine the numerators over the common denominator.

$\frac{- 7 {(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}} - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{1}}$

Step 24

Multiply ${(1 - 49 x^{2})}^{\frac{1}{2}}$ by ${(1 - 49 x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 24.1

Move ${(1 - 49 x^{2})}^{\frac{1}{2}}$ .

$\frac{- 7 ({(1 - 49 x^{2})}^{\frac{1}{2}} {(1 - 49 x^{2})}^{\frac{1}{2}}) - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{1}}$

Step 24.2

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

$\frac{- 7 {(1 - 49 x^{2})}^{\frac{1}{2} + \frac{1}{2}} - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{1}}$

Step 24.3

Combine the numerators over the common denominator.

$\frac{- 7 {(1 - 49 x^{2})}^{\frac{1 + 1}{2}} - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{1}}$

Step 24.4

Add $1$ and $1$ .

$\frac{- 7 {(1 - 49 x^{2})}^{\frac{2}{2}} - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{1}}$

Step 24.5

Divide $2$ by $2$ .

$\frac{- 7 {(1 - 49 x^{2})}^{1} - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{1}}$

Step 25

Simplify $- 7 {(1 - 49 x^{2})}^{1}$ .

$\frac{- 7 (1 - 49 x^{2}) - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{{(1 - 49 x^{2})}^{1}}$

Step 26

Simplify.

$\frac{- 7 (1 - 49 x^{2}) - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{1 - 49 x^{2}}$

Step 27

Simplify.

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Step 27.1

Apply the distributive property.

$\frac{- 7 \cdot 1 - 7 (- 49 x^{2}) - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{1 - 49 x^{2}}$

Step 27.2

Simplify the numerator.

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Step 27.2.1

Simplify each term.

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Step 27.2.1.1

Multiply $- 7$ by $1$ .

$\frac{- 7 - 7 (- 49 x^{2}) - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{1 - 49 x^{2}}$

Step 27.2.1.2

Multiply $- 49$ by $- 7$ .

$\frac{- 7 + 343 x^{2} - 49 \arccos (7 x) x \sqrt{1 - 49 x^{2}}}{1 - 49 x^{2}}$

Step 27.2.1.3

Rewrite $1$ as $1^{2}$ .

$\frac{- 7 + 343 x^{2} - 49 \arccos (7 x) x \sqrt{1^{2} - 49 x^{2}}}{1 - 49 x^{2}}$

Step 27.2.1.4

Rewrite $49 x^{2}$ as ${(7 x)}^{2}$ .

$\frac{- 7 + 343 x^{2} - 49 \arccos (7 x) x \sqrt{1^{2} - {(7 x)}^{2}}}{1 - 49 x^{2}}$

Step 27.2.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 7 x$ .

$\frac{- 7 + 343 x^{2} - 49 \arccos (7 x) x \sqrt{(1 + 7 x) (1 - (7 x))}}{1 - 49 x^{2}}$

Step 27.2.1.6

Multiply $7$ by $- 1$ .

$\frac{- 7 + 343 x^{2} - 49 \arccos (7 x) x \sqrt{(1 + 7 x) (1 - 7 x)}}{1 - 49 x^{2}}$

Step 27.2.2

Reorder factors in $- 7 + 343 x^{2} - 49 \arccos (7 x) x \sqrt{(1 + 7 x) (1 - 7 x)}$ .

$\frac{- 7 + 343 x^{2} - 49 x \arccos (7 x) \sqrt{(1 + 7 x) (1 - 7 x)}}{1 - 49 x^{2}}$

Step 27.3

Reorder terms.

$\frac{343 x^{2} - 49 x \sqrt{(1 + 7 x) (1 - 7 x)} \arccos (7 x) - 7}{- 49 x^{2} + 1}$