Calculus Examples

$\frac{5}{x^{3}} + 3 \sqrt[4]{y} = 18$

Step 1

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

$\frac{5}{x^{3}} + 3 y^{\frac{1}{4}} = 18$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (\frac{5}{x^{3}} + 3 y^{\frac{1}{4}}) = \frac{d}{d x} (18)$

Step 3

Differentiate the left side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $\frac{5}{x^{3}} + 3 y^{\frac{1}{4}}$ with respect to $x$ is $\frac{d}{d x} [\frac{5}{x^{3}}] + \frac{d}{d x} [3 y^{\frac{1}{4}}]$ .

$\frac{d}{d x} [\frac{5}{x^{3}}] + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{5}{x^{3}}]$ .

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Step 3.2.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x^{3}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$5 \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$5 \frac{d}{d x} [{(x^{3})}^{- 1}] + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.3

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

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{3}$ .

$5 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.3.2

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

$5 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $x^{3}$ .

$5 (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.4

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

$5 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.5

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

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Step 3.2.5.1

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

$5 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.5.2

Multiply $3$ by $- 2$ .

$5 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.6

Multiply $3$ by $- 1$ .

$5 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 3.2.7.1

Move $x^{2}$ .

$5 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.7.2

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

$5 (- 3 x^{2 - 6}) + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.7.3

Subtract $6$ from $2$ .

$5 (- 3 x^{- 4}) + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.2.8

Multiply $- 3$ by $5$ .

$- 15 x^{- 4} + \frac{d}{d x} [3 y^{\frac{1}{4}}]$

Step 3.3

Evaluate $\frac{d}{d x} [3 y^{\frac{1}{4}}]$ .

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Step 3.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 y^{\frac{1}{4}}$ with respect to $x$ is $3 \frac{d}{d x} [y^{\frac{1}{4}}]$ .

$- 15 x^{- 4} + 3 \frac{d}{d x} [y^{\frac{1}{4}}]$

Step 3.3.2

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

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Step 3.3.2.1

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

$- 15 x^{- 4} + 3 (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{4}}] \frac{d}{d x} [y])$

Step 3.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{4}$ .

$- 15 x^{- 4} + 3 (\frac{1}{4} {u_{2}}^{\frac{1}{4} - 1} \frac{d}{d x} [y])$

Step 3.3.2.3

Replace all occurrences of $u_{2}$ with $y$ .

$- 15 x^{- 4} + 3 (\frac{1}{4} y^{\frac{1}{4} - 1} \frac{d}{d x} [y])$

Step 3.3.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$- 15 x^{- 4} + 3 (\frac{1}{4} y^{\frac{1}{4} - 1} y')$

Step 3.3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 15 x^{- 4} + 3 (\frac{1}{4} y^{\frac{1}{4} - 1 \cdot \frac{4}{4}} y')$

Step 3.3.5

Combine $- 1$ and $\frac{4}{4}$ .

$- 15 x^{- 4} + 3 (\frac{1}{4} y^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}} y')$

Step 3.3.6

Combine the numerators over the common denominator.

$- 15 x^{- 4} + 3 (\frac{1}{4} y^{\frac{1 - 1 \cdot 4}{4}} y')$

Step 3.3.7

Simplify the numerator.

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Step 3.3.7.1

Multiply $- 1$ by $4$ .

$- 15 x^{- 4} + 3 (\frac{1}{4} y^{\frac{1 - 4}{4}} y')$

Step 3.3.7.2

Subtract $4$ from $1$ .

$- 15 x^{- 4} + 3 (\frac{1}{4} y^{\frac{- 3}{4}} y')$

Step 3.3.8

Move the negative in front of the fraction.

$- 15 x^{- 4} + 3 (\frac{1}{4} y^{- \frac{3}{4}} y')$

Step 3.3.9

Combine $\frac{1}{4}$ and $y^{- \frac{3}{4}}$ .

$- 15 x^{- 4} + 3 (\frac{y^{- \frac{3}{4}}}{4} y')$

Step 3.3.10

Combine $\frac{y^{- \frac{3}{4}}}{4}$ and $y'$ .

$- 15 x^{- 4} + 3 \frac{y^{- \frac{3}{4}} y'}{4}$

Step 3.3.11

Move $y^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 15 x^{- 4} + 3 \frac{y'}{4 y^{\frac{3}{4}}}$

Step 3.3.12

Combine $3$ and $\frac{y'}{4 y^{\frac{3}{4}}}$ .

$- 15 x^{- 4} + \frac{3 y'}{4 y^{\frac{3}{4}}}$

Step 3.4

Simplify.

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Step 3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 15 \frac{1}{x^{4}} + \frac{3 y'}{4 y^{\frac{3}{4}}}$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $- 15$ and $\frac{1}{x^{4}}$ .

$\frac{- 15}{x^{4}} + \frac{3 y'}{4 y^{\frac{3}{4}}}$

Step 3.4.2.2

Move the negative in front of the fraction.

$- \frac{15}{x^{4}} + \frac{3 y'}{4 y^{\frac{3}{4}}}$

Step 4

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$0$

Step 5

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

$- \frac{15}{x^{4}} + \frac{3 y'}{4 y^{\frac{3}{4}}} = 0$

Step 6

Solve for $y'$ .

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Step 6.1

Add $\frac{15}{x^{4}}$ to both sides of the equation.

$\frac{3 y'}{4 y^{\frac{3}{4}}} = \frac{15}{x^{4}}$

Step 6.2

Multiply both sides by $4 y^{\frac{3}{4}}$ .

$\frac{3 y'}{4 y^{\frac{3}{4}}} (4 y^{\frac{3}{4}}) = \frac{15}{x^{4}} (4 y^{\frac{3}{4}})$

Step 6.3

Simplify.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Simplify $\frac{3 y'}{4 y^{\frac{3}{4}}} (4 y^{\frac{3}{4}})$ .

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Step 6.3.1.1.1

Rewrite using the commutative property of multiplication.

$4 \frac{3 y'}{4 y^{\frac{3}{4}}} y^{\frac{3}{4}} = \frac{15}{x^{4}} (4 y^{\frac{3}{4}})$

Step 6.3.1.1.2

Cancel the common factor of $4$ .

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Step 6.3.1.1.2.1

Cancel the common factor.

$4 \frac{3 y'}{4 y^{\frac{3}{4}}} y^{\frac{3}{4}} = \frac{15}{x^{4}} (4 y^{\frac{3}{4}})$

Step 6.3.1.1.2.2

Rewrite the expression.

$\frac{3 y'}{y^{\frac{3}{4}}} y^{\frac{3}{4}} = \frac{15}{x^{4}} (4 y^{\frac{3}{4}})$

Step 6.3.1.1.3

Cancel the common factor of $y^{\frac{3}{4}}$ .

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Step 6.3.1.1.3.1

Cancel the common factor.

$\frac{3 y'}{y^{\frac{3}{4}}} y^{\frac{3}{4}} = \frac{15}{x^{4}} (4 y^{\frac{3}{4}})$

Step 6.3.1.1.3.2

Rewrite the expression.

$3 y' = \frac{15}{x^{4}} (4 y^{\frac{3}{4}})$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Simplify $\frac{15}{x^{4}} (4 y^{\frac{3}{4}})$ .

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Step 6.3.2.1.1

Rewrite using the commutative property of multiplication.

$3 y' = 4 \frac{15}{x^{4}} y^{\frac{3}{4}}$

Step 6.3.2.1.2

Multiply $4 \frac{15}{x^{4}}$ .

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Step 6.3.2.1.2.1

Combine $4$ and $\frac{15}{x^{4}}$ .

$3 y' = \frac{4 \cdot 15}{x^{4}} y^{\frac{3}{4}}$

Step 6.3.2.1.2.2

Multiply $4$ by $15$ .

$3 y' = \frac{60}{x^{4}} y^{\frac{3}{4}}$

Step 6.3.2.1.3

Combine $\frac{60}{x^{4}}$ and $y^{\frac{3}{4}}$ .

$3 y' = \frac{60 y^{\frac{3}{4}}}{x^{4}}$

Step 6.4

Divide each term in $3 y' = \frac{60 y^{\frac{3}{4}}}{x^{4}}$ by $3$ and simplify.

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Step 6.4.1

Divide each term in $3 y' = \frac{60 y^{\frac{3}{4}}}{x^{4}}$ by $3$ .

$\frac{3 y'}{3} = \frac{\frac{60 y^{\frac{3}{4}}}{x^{4}}}{3}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $3$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{3 y'}{3} = \frac{\frac{60 y^{\frac{3}{4}}}{x^{4}}}{3}$

Step 6.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{60 y^{\frac{3}{4}}}{x^{4}}}{3}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{60 y^{\frac{3}{4}}}{x^{4}} \cdot \frac{1}{3}$

Step 6.4.3.2

Combine.

$y' = \frac{60 y^{\frac{3}{4}} \cdot 1}{x^{4} \cdot 3}$

Step 6.4.3.3

Factor $3$ out of $60 y^{\frac{3}{4}} \cdot 1$ .

$y' = \frac{3 (20 y^{\frac{3}{4}} \cdot 1)}{x^{4} \cdot 3}$

Step 6.4.3.4

Cancel the common factors.

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Step 6.4.3.4.1

Factor $3$ out of $x^{4} \cdot 3$ .

$y' = \frac{3 (20 y^{\frac{3}{4}} \cdot 1)}{3 \cdot x^{4}}$

Step 6.4.3.4.2

Cancel the common factor.

$y' = \frac{3 (20 y^{\frac{3}{4}} \cdot 1)}{3 \cdot x^{4}}$

Step 6.4.3.4.3

Rewrite the expression.

$y' = \frac{20 y^{\frac{3}{4}} \cdot 1}{x^{4}}$

Step 6.4.3.5

Multiply $20$ by $1$ .

$y' = \frac{20 y^{\frac{3}{4}}}{x^{4}}$

Step 7

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

$\frac{d y}{d x} = \frac{20 y^{\frac{3}{4}}}{x^{4}}$