Calculus Examples

$y^{2} = \frac{x^{2} - 49}{x^{2} + 49}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2}) = \frac{d}{d x} (\frac{x^{2} - 49}{x^{2} + 49})$

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

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} - 49$ and $g (x) = x^{2} + 49$ .

$\frac{(x^{2} + 49) \frac{d}{d x} [x^{2} - 49] - (x^{2} - 49) \frac{d}{d x} [x^{2} + 49]}{{(x^{2} + 49)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $x^{2} - 49$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 49]$ .

$\frac{(x^{2} + 49) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 49]) - (x^{2} - 49) \frac{d}{d x} [x^{2} + 49]}{{(x^{2} + 49)}^{2}}$

Step 3.2.2

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

$\frac{(x^{2} + 49) (2 x + \frac{d}{d x} [- 49]) - (x^{2} - 49) \frac{d}{d x} [x^{2} + 49]}{{(x^{2} + 49)}^{2}}$

Step 3.2.3

Since $- 49$ is constant with respect to $x$ , the derivative of $- 49$ with respect to $x$ is $0$ .

$\frac{(x^{2} + 49) (2 x + 0) - (x^{2} - 49) \frac{d}{d x} [x^{2} + 49]}{{(x^{2} + 49)}^{2}}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $2 x$ and $0$ .

$\frac{(x^{2} + 49) (2 x) - (x^{2} - 49) \frac{d}{d x} [x^{2} + 49]}{{(x^{2} + 49)}^{2}}$

Step 3.2.4.2

Move $2$ to the left of $x^{2} + 49$ .

$\frac{2 \cdot (x^{2} + 49) x - (x^{2} - 49) \frac{d}{d x} [x^{2} + 49]}{{(x^{2} + 49)}^{2}}$

Step 3.2.5

By the Sum Rule, the derivative of $x^{2} + 49$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [49]$ .

$\frac{2 (x^{2} + 49) x - (x^{2} - 49) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [49])}{{(x^{2} + 49)}^{2}}$

Step 3.2.6

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

$\frac{2 (x^{2} + 49) x - (x^{2} - 49) (2 x + \frac{d}{d x} [49])}{{(x^{2} + 49)}^{2}}$

Step 3.2.7

Since $49$ is constant with respect to $x$ , the derivative of $49$ with respect to $x$ is $0$ .

$\frac{2 (x^{2} + 49) x - (x^{2} - 49) (2 x + 0)}{{(x^{2} + 49)}^{2}}$

Step 3.2.8

Simplify the expression.

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Step 3.2.8.1

Add $2 x$ and $0$ .

$\frac{2 (x^{2} + 49) x - (x^{2} - 49) (2 x)}{{(x^{2} + 49)}^{2}}$

Step 3.2.8.2

Multiply $2$ by $- 1$ .

$\frac{2 (x^{2} + 49) x - 2 (x^{2} - 49) x}{{(x^{2} + 49)}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{(2 x^{2} + 2 \cdot 49) x - 2 (x^{2} - 49) x}{{(x^{2} + 49)}^{2}}$

Step 3.3.2

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 49 x - 2 (x^{2} - 49) x}{{(x^{2} + 49)}^{2}}$

Step 3.3.3

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 49 x + (- 2 x^{2} - 2 \cdot - 49) x}{{(x^{2} + 49)}^{2}}$

Step 3.3.4

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 49 x - 2 x^{2} x - 2 \cdot - 49 x}{{(x^{2} + 49)}^{2}}$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Combine the opposite terms in $2 x^{2} x + 2 \cdot 49 x - 2 x^{2} x - 2 \cdot - 49 x$ .

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Step 3.3.5.1.1

Subtract $2 x^{2} x$ from $2 x^{2} x$ .

$\frac{2 \cdot 49 x + 0 - 2 \cdot - 49 x}{{(x^{2} + 49)}^{2}}$

Step 3.3.5.1.2

Add $2 \cdot 49 x$ and $0$ .

$\frac{2 \cdot 49 x - 2 \cdot - 49 x}{{(x^{2} + 49)}^{2}}$

Step 3.3.5.2

Simplify each term.

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Step 3.3.5.2.1

Multiply $2$ by $49$ .

$\frac{98 x - 2 \cdot - 49 x}{{(x^{2} + 49)}^{2}}$

Step 3.3.5.2.2

Multiply $- 2$ by $- 49$ .

$\frac{98 x + 98 x}{{(x^{2} + 49)}^{2}}$

Step 3.3.5.3

Add $98 x$ and $98 x$ .

$\frac{196 x}{{(x^{2} + 49)}^{2}}$

Step 4

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

$2 y y' = \frac{196 x}{{(x^{2} + 49)}^{2}}$

Step 5

Divide each term in $2 y y' = \frac{196 x}{{(x^{2} + 49)}^{2}}$ by $2 y$ and simplify.

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Step 5.1

Divide each term in $2 y y' = \frac{196 x}{{(x^{2} + 49)}^{2}}$ by $2 y$ .

$\frac{2 y y'}{2 y} = \frac{\frac{196 x}{{(x^{2} + 49)}^{2}}}{2 y}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 y y'}{2 y} = \frac{\frac{196 x}{{(x^{2} + 49)}^{2}}}{2 y}$

Step 5.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{\frac{196 x}{{(x^{2} + 49)}^{2}}}{2 y}$

Step 5.2.2

Cancel the common factor of $y$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{\frac{196 x}{{(x^{2} + 49)}^{2}}}{2 y}$

Step 5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{196 x}{{(x^{2} + 49)}^{2}}}{2 y}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{196 x}{{(x^{2} + 49)}^{2}} \cdot \frac{1}{2 y}$

Step 5.3.2

Combine.

$y' = \frac{196 x \cdot 1}{{(x^{2} + 49)}^{2} (2 y)}$

Step 5.3.3

Cancel the common factor of $196$ and $2$ .

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Step 5.3.3.1

Factor $2$ out of $196 x \cdot 1$ .

$y' = \frac{2 (98 x \cdot 1)}{{(x^{2} + 49)}^{2} (2 y)}$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $2$ out of ${(x^{2} + 49)}^{2} (2 y)$ .

$y' = \frac{2 (98 x \cdot 1)}{2 ({(x^{2} + 49)}^{2} (y))}$

Step 5.3.3.2.2

Cancel the common factor.

$y' = \frac{2 (98 x \cdot 1)}{2 ({(x^{2} + 49)}^{2} (y))}$

Step 5.3.3.2.3

Rewrite the expression.

$y' = \frac{98 x \cdot 1}{{(x^{2} + 49)}^{2} (y)}$

Step 5.3.4

Simplify the expression.

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Step 5.3.4.1

Multiply $98$ by $1$ .

$y' = \frac{98 x}{{(x^{2} + 49)}^{2} y}$

Step 5.3.4.2

Reorder factors in $\frac{98 x}{{(x^{2} + 49)}^{2} y}$ .

$y' = \frac{98 x}{y {(x^{2} + 49)}^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{98 x}{y {(x^{2} + 49)}^{2}}$