Calculus Examples

$x^{3} + y^{3} = 3 x y^{2}$

Step 1

Differentiate both sides of the equation.

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

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 $x^{3} + y^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [y^{3}]$ .

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

Step 2.1.2

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

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

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 $y$ .

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

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 $n = 3$ .

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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 $f (x) = x$ and $g (x) = y^{2}$ .

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

Step 3.3

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

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Step 3.3.1

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

$3 (x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 3.3.2

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

$3 (x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 3.3.3

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

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

Step 3.4

Move $2$ to the left of $x$ .

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

Step 3.5

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

$3 (2 x y y' + y^{2} \frac{d}{d x} [x])$

Step 3.6

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

$3 (2 x y y' + y^{2} \cdot 1)$

Step 3.7

Multiply $y^{2}$ by $1$ .

$3 (2 x y y' + y^{2})$

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

$3 (2 x y y') + 3 y^{2}$

Step 3.8.2

Multiply $2$ by $3$ .

$6 x y y' + 3 y^{2}$

Step 3.8.3

Reorder terms.

$3 y^{2} + 6 x y y'$

Step 4

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

$3 x^{2} + 3 y^{2} y' = 3 y^{2} + 6 x y y'$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $6 x y y'$ from both sides of the equation.

$3 x^{2} + 3 y^{2} y' - 6 x y y' = 3 y^{2}$

Step 5.2

Subtract $3 x^{2}$ from both sides of the equation.

$3 y^{2} y' - 6 x y y' = 3 y^{2} - 3 x^{2}$

Step 5.3

Factor $3 y y'$ out of $3 y^{2} y' - 6 x y y'$ .

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Step 5.3.1

Factor $3 y y'$ out of $3 y^{2} y'$ .

$3 y y' y - 6 x y y' = 3 y^{2} - 3 x^{2}$

Step 5.3.2

Factor $3 y y'$ out of $- 6 x y y'$ .

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

Step 5.3.3

Factor $3 y y'$ out of $3 y y' y + 3 y y' (- 2 x)$ .

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $3$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Rewrite the expression.

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

Step 5.4.2.2

Cancel the common factor of $y$ .

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Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Rewrite the expression.

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

Step 5.4.2.3

Cancel the common factor of $y - 2 x$ .

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Step 5.4.2.3.1

Cancel the common factor.

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

Step 5.4.2.3.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Cancel the common factor of $3$ .

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Step 5.4.3.1.1.1

Cancel the common factor.

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

Step 5.4.3.1.1.2

Rewrite the expression.

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

Step 5.4.3.1.2

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

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Step 5.4.3.1.2.1

Factor $y$ out of $y^{2}$ .

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

Step 5.4.3.1.2.2

Cancel the common factors.

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Step 5.4.3.1.2.2.1

Cancel the common factor.

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

Step 5.4.3.1.2.2.2

Rewrite the expression.

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

Step 5.4.3.1.3

Cancel the common factor of $- 3$ and $3$ .

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Step 5.4.3.1.3.1

Factor $3$ out of $- 3 x^{2}$ .

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

Step 5.4.3.1.3.2

Cancel the common factors.

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Step 5.4.3.1.3.2.1

Factor $3$ out of $3 y (y - 2 x)$ .

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

Step 5.4.3.1.3.2.2

Cancel the common factor.

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

Step 5.4.3.1.3.2.3

Rewrite the expression.

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

Step 5.4.3.1.4

Move the negative in front of the fraction.

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

Step 5.4.3.2

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

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

Step 5.4.3.3

Write each expression with a common denominator of $(y - 2 x) y$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.3.1

Multiply $\frac{y}{y - 2 x}$ by $\frac{y}{y}$ .

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

Step 5.4.3.3.2

Reorder the factors of $(y - 2 x) y$ .

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

Step 5.4.3.4

Combine the numerators over the common denominator.

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

Step 5.4.3.5

Simplify the numerator.

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Step 5.4.3.5.1

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

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

Step 5.4.3.5.2

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

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

Step 5.4.3.5.3

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

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

Step 5.4.3.5.4

Add $1$ and $1$ .

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

Step 5.4.3.5.5

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

$y' = \frac{(y + x) (y - x)}{y (y - 2 x)}$

Step 6

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

$\frac{d y}{d x} = \frac{(y + x) (y - x)}{y (y - 2 x)}$