Calculus Examples

$y = arcsec (\sqrt{3 x^{3}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x^{3}}$ as ${(3 x^{3})}^{\frac{1}{2}}$ .

$y = arcsec ({(3 x^{3})}^{\frac{1}{2}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (arcsec ({(3 x^{3})}^{\frac{1}{2}}))$

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

Simplify with factoring out.

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Step 4.1.1

Factor $3$ out of $3 x^{3}$ .

$\frac{d}{d x} [arcsec ({(3 (x^{3}))}^{\frac{1}{2}})]$

Step 4.1.2

Apply basic rules of exponents.

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Step 4.1.2.1

Apply the product rule to $3 (x^{3})$ .

$\frac{d}{d x} [arcsec (3^{\frac{1}{2}} {(x^{3})}^{\frac{1}{2}})]$

Step 4.1.2.2

Multiply the exponents in ${(x^{3})}^{\frac{1}{2}}$ .

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Step 4.1.2.2.1

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

$\frac{d}{d x} [arcsec (3^{\frac{1}{2}} x^{3 (\frac{1}{2})})]$

Step 4.1.2.2.2

Combine $3$ and $\frac{1}{2}$ .

$\frac{d}{d x} [arcsec (3^{\frac{1}{2}} x^{\frac{3}{2}})]$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = arcsec (x)$ and $g (x) = 3^{\frac{1}{2}} x^{\frac{3}{2}}$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $3^{\frac{1}{2}} x^{\frac{3}{2}}$ .

$\frac{d}{d u} [arcsec (u)] \frac{d}{d x} [3^{\frac{1}{2}} x^{\frac{3}{2}}]$

Step 4.2.2

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

$\frac{1}{u \sqrt{u^{2} - 1}} \frac{d}{d x} [3^{\frac{1}{2}} x^{\frac{3}{2}}]$

Step 4.2.3

Replace all occurrences of $u$ with $3^{\frac{1}{2}} x^{\frac{3}{2}}$ .

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Step 4.2.3.1

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

$\frac{1}{3^{\frac{1}{2}} x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{3 (\frac{1}{2})})}^{2} - 1}} \frac{d}{d x} [3^{\frac{1}{2}} x^{\frac{3}{2}}]$

Step 4.2.3.2

Combine $3$ and $\frac{1}{2}$ .

$\frac{1}{3^{\frac{1}{2}} x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} \frac{d}{d x} [3^{\frac{1}{2}} x^{\frac{3}{2}}]$

Step 4.3

Differentiate.

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Step 4.3.1

Since $3^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $3^{\frac{1}{2}} x^{\frac{3}{2}}$ with respect to $x$ is $3^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$\frac{1}{3^{\frac{1}{2}} x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} (3^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{3}{2}}])$

Step 4.3.2

Simplify terms.

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Step 4.3.2.1

Combine $3^{\frac{1}{2}}$ and $\frac{1}{3^{\frac{1}{2}} x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}}$ .

$\frac{3^{\frac{1}{2}}}{3^{\frac{1}{2}} x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 4.3.2.2

Cancel the common factor.

$\frac{3^{\frac{1}{2}}}{3^{\frac{1}{2}} x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 4.3.2.3

Rewrite the expression.

$\frac{1}{x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 4.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{1}{x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 4.4

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

$\frac{1}{x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 4.5

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

$\frac{1}{x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 4.6

Combine the numerators over the common denominator.

$\frac{1}{x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $2$ .

$\frac{1}{x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} (\frac{3}{2} x^{\frac{3 - 2}{2}})$

Step 4.7.2

Subtract $2$ from $3$ .

$\frac{1}{x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} (\frac{3}{2} x^{\frac{1}{2}})$

Step 4.8

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$\frac{1}{x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}} \cdot \frac{3 x^{\frac{1}{2}}}{2}$

Step 4.9

Multiply $\frac{1}{x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}}$ by $\frac{3 x^{\frac{1}{2}}}{2}$ .

$\frac{3 x^{\frac{1}{2}}}{x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1} \cdot 2}$

Step 4.10

Simplify the expression.

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Step 4.10.1

Move $2$ to the left of $x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}$ .

$\frac{3 x^{\frac{1}{2}}}{2 \cdot (x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1})}$

Step 4.10.2

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

$\frac{3}{2 x^{\frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1} x^{- \frac{1}{2}}}$

Step 4.11

Simplify the denominator.

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Step 4.11.1

Multiply $x^{\frac{3}{2}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.11.1.1

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

$\frac{3}{2 (x^{- \frac{1}{2}} x^{\frac{3}{2}}) \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.11.1.2

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

$\frac{3}{2 x^{- \frac{1}{2} + \frac{3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.11.1.3

Combine the numerators over the common denominator.

$\frac{3}{2 x^{\frac{- 1 + 3}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.11.1.4

Add $- 1$ and $3$ .

$\frac{3}{2 x^{\frac{2}{2}} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.11.1.5

Divide $2$ by $2$ .

$\frac{3}{2 x^{1} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.11.2

Simplify $2 x^{1} \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}$ .

$\frac{3}{2 x \sqrt{{(3^{\frac{1}{2}} x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.12

Simplify.

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Step 4.12.1

Apply the product rule to $3^{\frac{1}{2}} x^{\frac{3}{2}}$ .

$\frac{3}{2 x \sqrt{{(3^{\frac{1}{2}})}^{2} {(x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.12.2

Combine terms.

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Step 4.12.2.1

Multiply the exponents in ${(3^{\frac{1}{2}})}^{2}$ .

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Step 4.12.2.1.1

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

$\frac{3}{2 x \sqrt{3^{\frac{1}{2} \cdot 2} {(x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.12.2.1.2

Cancel the common factor of $2$ .

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Step 4.12.2.1.2.1

Cancel the common factor.

$\frac{3}{2 x \sqrt{3^{\frac{1}{2} \cdot 2} {(x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.12.2.1.2.2

Rewrite the expression.

$\frac{3}{2 x \sqrt{3^{1} {(x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.12.2.2

Evaluate the exponent.

$\frac{3}{2 x \sqrt{3 {(x^{\frac{3}{2}})}^{2} - 1}}$

Step 4.12.2.3

Multiply the exponents in ${(x^{\frac{3}{2}})}^{2}$ .

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Step 4.12.2.3.1

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

$\frac{3}{2 x \sqrt{3 x^{\frac{3}{2} \cdot 2} - 1}}$

Step 4.12.2.3.2

Cancel the common factor of $2$ .

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Step 4.12.2.3.2.1

Cancel the common factor.

$\frac{3}{2 x \sqrt{3 x^{\frac{3}{2} \cdot 2} - 1}}$

Step 4.12.2.3.2.2

Rewrite the expression.

$\frac{3}{2 x \sqrt{3 x^{3} - 1}}$

Step 4.12.3

Multiply $\frac{3}{2 x \sqrt{3 x^{3} - 1}}$ by $\frac{\sqrt{3 x^{3} - 1}}{\sqrt{3 x^{3} - 1}}$ .

$\frac{3}{2 x \sqrt{3 x^{3} - 1}} \cdot \frac{\sqrt{3 x^{3} - 1}}{\sqrt{3 x^{3} - 1}}$

Step 4.12.4

Combine and simplify the denominator.

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Step 4.12.4.1

Multiply $\frac{3}{2 x \sqrt{3 x^{3} - 1}}$ by $\frac{\sqrt{3 x^{3} - 1}}{\sqrt{3 x^{3} - 1}}$ .

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x \sqrt{3 x^{3} - 1} \sqrt{3 x^{3} - 1}}$

Step 4.12.4.2

Move $\sqrt{3 x^{3} - 1}$ .

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x (\sqrt{3 x^{3} - 1} \sqrt{3 x^{3} - 1})}$

Step 4.12.4.3

Raise $\sqrt{3 x^{3} - 1}$ to the power of $1$ .

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x ({\sqrt{3 x^{3} - 1}}^{1} \sqrt{3 x^{3} - 1})}$

Step 4.12.4.4

Raise $\sqrt{3 x^{3} - 1}$ to the power of $1$ .

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x ({\sqrt{3 x^{3} - 1}}^{1} {\sqrt{3 x^{3} - 1}}^{1})}$

Step 4.12.4.5

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

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x {\sqrt{3 x^{3} - 1}}^{1 + 1}}$

Step 4.12.4.6

Add $1$ and $1$ .

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x {\sqrt{3 x^{3} - 1}}^{2}}$

Step 4.12.4.7

Rewrite ${\sqrt{3 x^{3} - 1}}^{2}$ as $3 x^{3} - 1$ .

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Step 4.12.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x^{3} - 1}$ as ${(3 x^{3} - 1)}^{\frac{1}{2}}$ .

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x {({(3 x^{3} - 1)}^{\frac{1}{2}})}^{2}}$

Step 4.12.4.7.2

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

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x {(3 x^{3} - 1)}^{\frac{1}{2} \cdot 2}}$

Step 4.12.4.7.3

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

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x {(3 x^{3} - 1)}^{\frac{2}{2}}}$

Step 4.12.4.7.4

Cancel the common factor of $2$ .

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Step 4.12.4.7.4.1

Cancel the common factor.

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x {(3 x^{3} - 1)}^{\frac{2}{2}}}$

Step 4.12.4.7.4.2

Rewrite the expression.

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x {(3 x^{3} - 1)}^{1}}$

Step 4.12.4.7.5

Simplify.

$\frac{3 \sqrt{3 x^{3} - 1}}{2 x (3 x^{3} - 1)}$

Step 5

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

$y' = \frac{3 \sqrt{3 x^{3} - 1}}{2 x (3 x^{3} - 1)}$

Step 6

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

$\frac{d y}{d x} = \frac{3 \sqrt{3 x^{3} - 1}}{2 x (3 x^{3} - 1)}$