Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

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

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

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 $n = 3$ .

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $3$ by $- 1$ .

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

Step 2.4

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

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

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

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

$2 (x \cdot 1 + y \frac{d}{d y} [x])$

Step 3.3.2

Multiply $x$ by $1$ .

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

Step 3.4

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

$2 (x + y x')$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$2 x + 2 (y x')$

Step 3.5.2

Reorder terms.

$2 y x' + 2 x$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Subtract $2 y x'$ from both sides of the equation.

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

Step 5.2

Add $3 y^{2}$ to both sides of the equation.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Factor $x'$ out of $- 2 y x'$ .

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

Step 5.3.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $3 x^{2} - 2 y$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $x'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

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

Step 6

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

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