Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

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 $f (x) = x + 1$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [x + 1] - (x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{x (\frac{d}{d x} [x] + \frac{d}{d x} [1]) - (x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.2

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

$\frac{x (1 + \frac{d}{d x} [1]) - (x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.3

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

$\frac{x (1 + 0) - (x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

$\frac{x \cdot 1 - (x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.4.2

Multiply $x$ by $1$ .

$\frac{x - (x + 1) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.5

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

$\frac{x - (x + 1) \cdot 1}{x^{2}}$

Step 3.2.6

Multiply $- 1$ by $1$ .

$\frac{x - (x + 1)}{x^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{x - x - 1 \cdot 1}{x^{2}}$

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Subtract $x$ from $x$ .

$\frac{0 - 1 \cdot 1}{x^{2}}$

Step 3.3.2.2

Subtract $1 \cdot 1$ from $0$ .

$\frac{- 1 \cdot 1}{x^{2}}$

Step 3.3.2.3

Multiply $- 1$ by $1$ .

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

Step 3.3.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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