Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

$2 x + 2 y y'$

Step 2.3

Reorder terms.

$2 y y' + 2 x$

Step 3

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

$y'$

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.

$2 y y' + 2 x - y' = 0$

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

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

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Step 5.4.1

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 5.4.2

Simplify the left side.

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Step 5.4.2.1

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

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Move the negative in front of the fraction.

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

Step 6

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

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