Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} - y^{3}) = \frac{d}{d x} (6 x y)$

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 $x^{3} - y^{3}$ 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

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $3$ by $- 1$ .

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x y$ with respect to $x$ is $6 \frac{d}{d x} [x y]$ .

$6 \frac{d}{d x} [x y]$

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 $g (x) = y$ .

$6 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 3.3

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

$6 (x y' + y \frac{d}{d x} [x])$

Step 3.4

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

$6 (x y' + y \cdot 1)$

Step 3.5

Multiply $y$ by $1$ .

$6 (x y' + y)$

Step 3.6

Apply the distributive property.

$6 x y' + 6 y$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

Divide each term in $3 y' (- y^{2} - 2 x) = 6 y - 3 x^{2}$ by $3 (- y^{2} - 2 x)$ and simplify.

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Step 5.4.1

Divide each term in $3 y' (- y^{2} - 2 x) = 6 y - 3 x^{2}$ by $3 (- y^{2} - 2 x)$ .

$\frac{3 y' (- y^{2} - 2 x)}{3 (- y^{2} - 2 x)} = \frac{6 y}{3 (- y^{2} - 2 x)} + \frac{- 3 x^{2}}{3 (- y^{2} - 2 x)}$

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.

$\frac{3 y' (- y^{2} - 2 x)}{3 (- y^{2} - 2 x)} = \frac{6 y}{3 (- y^{2} - 2 x)} + \frac{- 3 x^{2}}{3 (- y^{2} - 2 x)}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{y' (- y^{2} - 2 x)}{- y^{2} - 2 x} = \frac{6 y}{3 (- y^{2} - 2 x)} + \frac{- 3 x^{2}}{3 (- y^{2} - 2 x)}$

Step 5.4.2.2

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

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{y' (- y^{2} - 2 x)}{- y^{2} - 2 x} = \frac{6 y}{3 (- y^{2} - 2 x)} + \frac{- 3 x^{2}}{3 (- y^{2} - 2 x)}$

Step 5.4.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{6 y}{3 (- y^{2} - 2 x)} + \frac{- 3 x^{2}}{3 (- y^{2} - 2 x)}$

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 $6$ and $3$ .

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Step 5.4.3.1.1.1

Factor $3$ out of $6 y$ .

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

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Cancel the common factor.

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

Step 5.4.3.1.1.2.2

Rewrite the expression.

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

Step 5.4.3.1.2

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

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Step 5.4.3.1.2.1

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

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

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.

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

Step 5.4.3.1.2.2.2

Rewrite the expression.

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

Step 5.4.3.1.3

Move the negative in front of the fraction.

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

Step 5.4.3.2

Simplify terms.

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Step 5.4.3.2.1

Combine the numerators over the common denominator.

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

Step 5.4.3.2.2

Factor $- 1$ out of $- y^{2}$ .

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

Step 5.4.3.2.3

Factor $- 1$ out of $- 2 x$ .

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

Step 5.4.3.2.4

Factor $- 1$ out of $- (y^{2}) - (2 x)$ .

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

Step 5.4.3.2.5

Rewrite negatives.

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Step 5.4.3.2.5.1

Rewrite $- (y^{2} + 2 x)$ as $- 1 (y^{2} + 2 x)$ .

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

Step 5.4.3.2.5.2

Move the negative in front of the fraction.

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

Step 6

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

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