Calculus Examples

$x y^{2} = y$ , $(1, 1)$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

Differentiate using the Power Rule.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 3

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

$1$

Step 4

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

$2 x y + y^{2} x' = 1$

Step 5

Solve for $x'$ .

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Step 5.1

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

$y^{2} x' = 1 - 2 x y$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

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

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Step 5.2.3.1.1.1

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

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

Step 5.2.3.1.1.2

Cancel the common factors.

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Step 5.2.3.1.1.2.1

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

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

Step 5.2.3.1.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.1.2.3

Rewrite the expression.

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

Step 5.2.3.1.2

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Replace $x$ with $1$ and $y$ with $1$ in the expression.

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

Step 8

Simplify the result.

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Step 8.1

Simplify each term.

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Step 8.1.1

One to any power is one.

$\frac{1}{1} - \frac{2 (1)}{1}$

Step 8.1.2

Divide $1$ by $1$ .

$1 - \frac{2 (1)}{1}$

Step 8.1.3

Divide $2 (1)$ by $1$ .

$1 - (2 (1))$

Step 8.1.4

Multiply $- (2 (1))$ .

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Step 8.1.4.1

Multiply $2$ by $1$ .

$1 - 1 \cdot 2$

Step 8.1.4.2

Multiply $- 1$ by $2$ .

$1 - 2$

Step 8.2

Subtract $2$ from $1$ .

$- 1$