Calculus Examples

$x^{2} - y^{2} = 9$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x^{2} - y^{2}) = \frac{d}{d y} (9)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = x$ .

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Step 2.2.1.1

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

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

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 $n = 2$ .

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

Since $- 1$ is constant with respect to $y$ , the derivative of $- y^{2}$ with respect to $y$ is $- \frac{d}{d y} [y^{2}]$ .

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

Step 2.3.2

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

$2 x x' - (2 y)$

Step 2.3.3

Multiply $2$ by $- 1$ .

$2 x x' - 2 y$

Step 3

Since $9$ is constant with respect to $y$ , the derivative of $9$ with respect to $y$ is $0$ .

$0$

Step 4

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

$2 x x' - 2 y = 0$

Step 5

Solve for $x'$ .

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Step 5.1

Add $2 y$ to both sides of the equation.

$2 x x' = 2 y$

Step 5.2

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

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Step 5.2.1

Divide each term in $2 x x' = 2 y$ by $2 x$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

Cancel the common factor of $x$ .

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide $x'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of $2$ .

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Step 5.2.3.1.1

Cancel the common factor.

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

Step 5.2.3.1.2

Rewrite the expression.

$x' = \frac{y}{x}$

Step 6

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

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