Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

By the Sum Rule, the derivative of $3 x^{2} y - 3 x y^{2}$ with respect to $x$ is $\frac{d}{d x} [3 x^{2} y] + \frac{d}{d x} [- 3 x y^{2}]$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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 3.3.3.1

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

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

Step 3.3.3.2

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

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

Step 3.3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

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

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

Step 3.3.7

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

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

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Apply the distributive property.

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

Step 3.4.3

Combine terms.

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Step 3.4.3.1

Multiply $2$ by $3$ .

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

Step 3.4.3.2

Multiply $2$ by $- 3$ .

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

Step 3.4.4

Reorder terms.

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 5.2

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

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

Step 5.3

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

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

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

Step 5.4.5

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

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

Step 5.5

Factor using the perfect square rule.

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Step 5.5.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x y = 2 \cdot x \cdot y$

Step 5.5.2

Rewrite the polynomial.

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

Step 5.5.3

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = y$ .

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $3$ .

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Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Rewrite the expression.

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

Step 5.6.2.2

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

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Step 5.6.2.2.1

Cancel the common factor.

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

Step 5.6.2.2.2

Divide $y'$ by $1$ .

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

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Simplify each term.

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Step 5.6.3.1.1

Cancel the common factor of $3$ .

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Step 5.6.3.1.1.1

Cancel the common factor.

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

Step 5.6.3.1.1.2

Rewrite the expression.

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

Step 5.6.3.1.2

Cancel the common factor of $3$ .

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Step 5.6.3.1.2.1

Cancel the common factor.

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

Step 5.6.3.1.2.2

Rewrite the expression.

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

Step 5.6.3.1.3

Cancel the common factor of $- 6$ and $3$ .

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Step 5.6.3.1.3.1

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

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

Step 5.6.3.1.3.2

Cancel the common factors.

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Step 5.6.3.1.3.2.1

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

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

Step 5.6.3.1.3.2.2

Cancel the common factor.

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

Step 5.6.3.1.3.2.3

Rewrite the expression.

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

Step 5.6.3.1.4

Move the negative in front of the fraction.

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

Step 6

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

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