Calculus Examples

${(x - y - 1)}^{3} = x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(x - y - 1)}^{3}) = \frac{d}{d x} (x)$

Step 2

Differentiate the left side of the equation.

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Step 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 $g (x) = x - y - 1$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x - y - 1$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [x - y - 1]$

Step 2.1.2

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

$3 u^{2} \frac{d}{d x} [x - y - 1]$

Step 2.1.3

Replace all occurrences of $u$ with $x - y - 1$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $x - y - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- y] + \frac{d}{d x} [- 1]$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$3 {(x - y - 1)}^{2} (1 - y' + 0)$

Step 2.5

Add $1 - y'$ and $0$ .

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Combine terms.

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Step 2.6.2.1

Multiply $3$ by $1$ .

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

Step 2.6.2.2

Multiply $- 1$ by $3$ .

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

Step 3

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

$1$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Reorder factors in $3 {(x - y - 1)}^{2} - 3 {(x - y - 1)}^{2} y'$ .

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

Step 5.2

Subtract $3 {(x - y - 1)}^{2}$ from both sides of the equation.

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

Step 5.3

Divide each term in $- 3 y' {(x - y - 1)}^{2} = 1 - 3 {(x - y - 1)}^{2}$ by $- 3 {(x - y - 1)}^{2}$ and simplify.

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Step 5.3.1

Divide each term in $- 3 y' {(x - y - 1)}^{2} = 1 - 3 {(x - y - 1)}^{2}$ by $- 3 {(x - y - 1)}^{2}$ .

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 3$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of ${(x - y - 1)}^{2}$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

To write $\frac{1}{- 3 {(x - y - 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Step 5.3.3.2

To write $\frac{- 3 {(x - y - 1)}^{2}}{- 3 {(x - y - 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Step 5.3.3.3

Write each expression with a common denominator of $3 {(x - y - 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.3.1

Multiply $\frac{1}{- 3 {(x - y - 1)}^{2}}$ by $\frac{- 1}{- 1}$ .

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

Step 5.3.3.3.2

Multiply $- 1$ by $- 3$ .

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

Step 5.3.3.3.3

Multiply $\frac{- 3 {(x - y - 1)}^{2}}{- 3 {(x - y - 1)}^{2}}$ by $\frac{- 1}{- 1}$ .

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

Step 5.3.3.3.4

Multiply $- 1$ by $- 3$ .

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

Step 5.3.3.4

Combine the numerators over the common denominator.

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

Step 5.3.3.5

Multiply $- 1$ by $- 3$ .

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

Step 5.3.3.6

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

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

Step 5.3.3.7

Factor $- 1$ out of $3 {(x - y - 1)}^{2}$ .

$y' = \frac{- 1 (1) - (- 3 {(x - y - 1)}^{2})}{3 {(x - y - 1)}^{2}}$

Step 5.3.3.8

Factor $- 1$ out of $- 1 (1) - (- 3 {(x - y - 1)}^{2})$ .

$y' = \frac{- 1 (1 - 3 {(x - y - 1)}^{2})}{3 {(x - y - 1)}^{2}}$

Step 5.3.3.9

Move the negative in front of the fraction.

$y' = - \frac{1 - 3 {(x - y - 1)}^{2}}{3 {(x - y - 1)}^{2}}$

Step 6

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

$\frac{d y}{d x} = - \frac{1 - 3 {(x - y - 1)}^{2}}{3 {(x - y - 1)}^{2}}$