Calculus Examples

$x^{3} - 3 x^{2} y + 3 x y^{2} - y^{3} = 10$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} - 3 x^{2} y + 3 x y^{2} - y^{3}) = \frac{d}{d x} (10)$

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 $x^{3} - 3 x^{2} y + 3 x y^{2} - y^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2} y] + \frac{d}{d x} [3 x y^{2}] + \frac{d}{d x} [- y^{3}]$ .

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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 $g (x) = y$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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 2.3.3.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 3 (x (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x]) + \frac{d}{d x} [- y^{3}]$

Step 2.3.3.2

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

$3 x^{2} - 3 (x^{2} y' + 2 y x) + 3 (x (2 u_{1} \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x]) + \frac{d}{d x} [- y^{3}]$

Step 2.3.3.3

Replace all occurrences of $u_{1}$ with $y$ .

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

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

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

Step 2.3.7

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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 2.4.2.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

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

Step 2.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

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

Step 2.4.2.3

Replace all occurrences of $u_{2}$ with $y$ .

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

Step 2.4.3

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

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

Step 2.4.4

Multiply $3$ by $- 1$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Combine terms.

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Step 2.5.3.1

Multiply $2$ by $- 3$ .

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

Step 2.5.3.2

Multiply $2$ by $3$ .

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

Step 2.5.4

Reorder terms.

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

Step 3

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$0$

Step 4

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

$3 x^{2} + 3 y^{2} - 3 x^{2} y' - 6 x y + 6 x y y' - 3 y^{2} y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Subtract $3 x^{2}$ from 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 5.1.2

Subtract $3 y^{2}$ from 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 5.1.3

Add $6 x y$ to 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 5.2

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

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Step 5.2.1

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 5.2.2

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 5.2.3

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 5.2.4

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 5.2.5

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 5.3

Factor.

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Step 5.3.1

Factor by grouping.

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Step 5.3.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = 2$ .

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Step 5.3.1.1.1

Reorder terms.

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

Step 5.3.1.1.2

Reorder $- y^{2}$ and $2 x y$ .

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

Step 5.3.1.1.3

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

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

Step 5.3.1.1.4

Rewrite $2$ as $1$ plus $1$

$3 y' (- x^{2} + (1 + 1) (x y) - y^{2}) = - 3 x^{2} - 3 y^{2} + 6 x y$

Step 5.3.1.1.5

Apply the distributive property.

$3 y' (- x^{2} + 1 (x y) + 1 (x y) - y^{2}) = - 3 x^{2} - 3 y^{2} + 6 x y$

Step 5.3.1.1.6

Multiply $x y$ by $1$ .

$3 y' (- x^{2} + x y + 1 (x y) - y^{2}) = - 3 x^{2} - 3 y^{2} + 6 x y$

Step 5.3.1.1.7

Multiply $x y$ by $1$ .

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

Step 5.3.1.2

Factor out the greatest common factor from each group.

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Step 5.3.1.2.1

Group the first two terms and the last two terms.

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

Step 5.3.1.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 5.3.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + y$ .

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

Step 5.3.2

Remove unnecessary parentheses.

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

Step 5.4

Combine exponents.

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Step 5.4.1

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

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

Step 5.4.2

Factor $- 1$ out of $y$ .

$3 y' (- (x) - 1 (- y)) (x - y) = - 3 x^{2} - 3 y^{2} + 6 x y$

Step 5.4.3

Factor $- 1$ out of $- (x) - 1 (- y)$ .

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

Step 5.4.4

Rewrite $- (x - y)$ as $- 1 (x - y)$ .

$3 y' (- 1 (x - y)) (x - y) = - 3 x^{2} - 3 y^{2} + 6 x y$

Step 5.4.5

Remove parentheses.

$3 y' \cdot (- 1 (x - y) (x - y)) = - 3 x^{2} - 3 y^{2} + 6 x y$

Step 5.4.6

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

$3 y' \cdot (- 1 ((x - y) (x - y))) = - 3 x^{2} - 3 y^{2} + 6 x y$

Step 5.4.7

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

$3 y' \cdot (- 1 ((x - y) (x - y))) = - 3 x^{2} - 3 y^{2} + 6 x y$

Step 5.4.8

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

$3 y' \cdot (- 1 {(x - y)}^{1 + 1}) = - 3 x^{2} - 3 y^{2} + 6 x y$

Step 5.4.9

Add $1$ and $1$ .

$3 y' \cdot (- 1 {(x - y)}^{2}) = - 3 x^{2} - 3 y^{2} + 6 x y$

Step 5.4.10

Multiply $- 1$ by $3$ .

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

Step 5.5

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 5.5.1

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 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $- 3$ .

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Step 5.5.2.1.1

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 5.5.2.1.2

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 5.5.2.2

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

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Step 5.5.2.2.1

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 5.5.2.2.2

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 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Cancel the common factor of $- 3$ .

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Step 5.5.3.1.1.1

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 5.5.3.1.1.2

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 5.5.3.1.2

Cancel the common factor of $- 3$ .

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Step 5.5.3.1.2.1

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 5.5.3.1.2.2

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 5.5.3.1.3

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

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Step 5.5.3.1.3.1

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 5.5.3.1.3.2

Cancel the common factors.

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Step 5.5.3.1.3.2.1

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 5.5.3.1.3.2.2

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 5.5.3.1.3.2.3

Rewrite the expression.

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

Step 5.5.3.1.4

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 $\frac{d y}{d x}$ .

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