Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

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

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 2.2.1.2

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

$5 x^{4} + 3 u^{2} \frac{d}{d x} [y]$

Step 2.2.1.3

Replace all occurrences of $u$ with $y$ .

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

Step 2.2.2

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

$5 x^{4} + 3 y^{2} y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$\frac{y \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [y]}{y^{2}}$

Step 3.2

Differentiate using the Constant Rule.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{y \frac{d}{d x} [1] - \frac{d}{d x} [y]}{y^{2}}$

Step 3.2.2

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

$\frac{y \cdot 0 - \frac{d}{d x} [y]}{y^{2}}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Multiply $y$ by $0$ .

$\frac{0 - \frac{d}{d x} [y]}{y^{2}}$

Step 3.2.3.2

Subtract $\frac{d}{d x} [y]$ from $0$ .

$\frac{- \frac{d}{d x} [y]}{y^{2}}$

Step 3.2.3.3

Move the negative in front of the fraction.

$- \frac{\frac{d}{d x} [y]}{y^{2}}$

Step 3.3

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

$- \frac{y'}{y^{2}}$

Step 4

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

$5 x^{4} + 3 y^{2} y' = - \frac{y'}{y^{2}}$

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $y^{2}$ .

$(5 x^{4} + 3 y^{2} y') y^{2} = - \frac{y'}{y^{2}} y^{2}$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $(5 x^{4} + 3 y^{2} y') y^{2}$ .

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Step 5.2.1.1.1

Apply the distributive property.

$5 x^{4} y^{2} + 3 y^{2} y' y^{2} = - \frac{y'}{y^{2}} y^{2}$

Step 5.2.1.1.2

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 5.2.1.1.2.1

Move $y^{2}$ .

$5 x^{4} y^{2} + 3 (y^{2} y^{2}) y' = - \frac{y'}{y^{2}} y^{2}$

Step 5.2.1.1.2.2

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

$5 x^{4} y^{2} + 3 y^{2 + 2} y' = - \frac{y'}{y^{2}} y^{2}$

Step 5.2.1.1.2.3

Add $2$ and $2$ .

$5 x^{4} y^{2} + 3 y^{4} y' = - \frac{y'}{y^{2}} y^{2}$

Step 5.2.1.1.3

Move $y^{4}$ .

$5 x^{4} y^{2} + 3 y' y^{4} = - \frac{y'}{y^{2}} y^{2}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Cancel the common factor of $y^{2}$ .

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Step 5.2.2.1.1

Move the leading negative in $- \frac{y'}{y^{2}}$ into the numerator.

$5 x^{4} y^{2} + 3 y' y^{4} = \frac{- y'}{y^{2}} y^{2}$

Step 5.2.2.1.2

Cancel the common factor.

$5 x^{4} y^{2} + 3 y' y^{4} = \frac{- y'}{y^{2}} y^{2}$

Step 5.2.2.1.3

Rewrite the expression.

$5 x^{4} y^{2} + 3 y' y^{4} = - y'$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Add $y'$ to both sides of the equation.

$5 x^{4} y^{2} + 3 y' y^{4} + y' = 0$

Step 5.3.2

Subtract $5 x^{4} y^{2}$ from both sides of the equation.

$3 y' y^{4} + y' = - 5 x^{4} y^{2}$

Step 5.3.3

Factor $y'$ out of $3 y' y^{4} + y'$ .

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Step 5.3.3.1

Factor $y'$ out of $3 y' y^{4}$ .

$y' (3 y^{4}) + y' = - 5 x^{4} y^{2}$

Step 5.3.3.2

Raise $y'$ to the power of $1$ .

$y' (3 y^{4}) + y' = - 5 x^{4} y^{2}$

Step 5.3.3.3

Factor $y'$ out of ${y'}^{1}$ .

$y' (3 y^{4}) + y' \cdot 1 = - 5 x^{4} y^{2}$

Step 5.3.3.4

Factor $y'$ out of $y' (3 y^{4}) + y' \cdot 1$ .

$y' (3 y^{4} + 1) = - 5 x^{4} y^{2}$

Step 5.3.4

Divide each term in $y' (3 y^{4} + 1) = - 5 x^{4} y^{2}$ by $3 y^{4} + 1$ and simplify.

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Step 5.3.4.1

Divide each term in $y' (3 y^{4} + 1) = - 5 x^{4} y^{2}$ by $3 y^{4} + 1$ .

$\frac{y' (3 y^{4} + 1)}{3 y^{4} + 1} = \frac{- 5 x^{4} y^{2}}{3 y^{4} + 1}$

Step 5.3.4.2

Simplify the left side.

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Step 5.3.4.2.1

Cancel the common factor of $3 y^{4} + 1$ .

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Step 5.3.4.2.1.1

Cancel the common factor.

$\frac{y' (3 y^{4} + 1)}{3 y^{4} + 1} = \frac{- 5 x^{4} y^{2}}{3 y^{4} + 1}$

Step 5.3.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 5 x^{4} y^{2}}{3 y^{4} + 1}$

Step 5.3.4.3

Simplify the right side.

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Step 5.3.4.3.1

Move the negative in front of the fraction.

$y' = - \frac{5 x^{4} y^{2}}{3 y^{4} + 1}$

Step 6

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

$\frac{d y}{d x} = - \frac{5 x^{4} y^{2}}{3 y^{4} + 1}$