Calculus Examples

$y = \frac{1}{x}$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = 1$ and $g (y) = x$ .

$\frac{x \frac{d}{d y} [1] - 1 \cdot 1 \frac{d}{d y} [x]}{x^{2}}$

Step 3.2

Differentiate using the Constant Rule.

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Step 3.2.1

Multiply $- 1$ by $1$ .

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

Step 3.2.2

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$\frac{x \cdot 0 - \frac{d}{d y} [x]}{x^{2}}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Multiply $x$ by $0$ .

$\frac{0 - \frac{d}{d y} [x]}{x^{2}}$

Step 3.2.3.2

Subtract $\frac{d}{d y} [x]$ from $0$ .

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

Step 3.2.3.3

Move the negative in front of the fraction.

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

Step 3.3

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

$- \frac{x'}{x^{2}}$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $- \frac{x'}{x^{2}} = 1$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

Divide $\frac{x'}{x^{2}}$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $1$ by $- 1$ .

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

Step 5.3

Multiply both sides by $x^{2}$ .

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

Step 5.4

Simplify the left side.

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Step 5.4.1

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

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Step 5.4.1.1

Cancel the common factor.

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

Step 5.4.1.2

Rewrite the expression.

$x' = - x^{2}$

Step 6

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

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