Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

$2 y y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$2 x + y'$

Step 4

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

$2 y y' = 2 x + y'$

Step 5

Solve for $y'$ .

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Step 5.1

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

$2 y y' - y' = 2 x$

Step 5.2

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

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Step 5.2.1

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

$y' (2 y) - y' = 2 x$

Step 5.2.2

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

$y' (2 y) + y' \cdot - 1 = 2 x$

Step 5.2.3

Factor $y'$ out of $y' (2 y) + y' \cdot - 1$ .

$y' (2 y - 1) = 2 x$

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2 y - 1$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 6

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

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

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Set the numerator equal to zero.

$2 x = 0$

Step 7.2

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 7.2.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $0$ by $2$ .

$x = 0$

Step 8

Solve for $y$ .

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Step 8.1

Simplify $0^{2} + y$ .

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Step 8.1.1

Raising $0$ to any positive power yields $0$ .

$y^{2} = 0 + y$

Step 8.1.2

Add $0$ and $y$ .

$y^{2} = y$

Step 8.2

Subtract $y$ from both sides of the equation.

$y^{2} - y = 0$

Step 8.3

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

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Step 8.3.1

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

$y \cdot y - y = 0$

Step 8.3.2

Factor $y$ out of $- y$ .

$y \cdot y + y \cdot - 1 = 0$

Step 8.3.3

Factor $y$ out of $y \cdot y + y \cdot - 1$ .

$y (y - 1) = 0$

Step 8.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y = 0$

$y - 1 = 0$

Step 8.5

Set $y$ equal to $0$ .

$y = 0$

Step 8.6

Set $y - 1$ equal to $0$ and solve for $y$ .

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Step 8.6.1

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

Step 8.6.2

Add $1$ to both sides of the equation.

$y = 1$

Step 8.7

The final solution is all the values that make $y (y - 1) = 0$ true.

$y = 0, 1$

Step 9

Find the points where $\frac{d y}{d x} = 0$ .

$(0, 0)$

$(0, 1)$

Step 10