Calculus Examples

$z = \frac{y}{y - x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (z) = \frac{d}{d x} (\frac{y}{y - x})$

Step 2

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

$z'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

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

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

Step 3.1.2

Rewrite $\frac{1}{y - x}$ as ${(y - x)}^{- 1}$ .

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

Step 3.2

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

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Step 3.2.1

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

$y (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [y - x])$

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 3.3.4

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

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

Step 3.3.5

Multiply.

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Step 3.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.5.2

Multiply ${(y - x)}^{- 2}$ by $1$ .

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

Step 3.3.6

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

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

Step 3.3.7

Multiply ${(y - x)}^{- 2}$ by $1$ .

$y {(y - x)}^{- 2}$

Step 3.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.5

Simplify.

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Step 3.5.1

Combine $y$ and $\frac{1}{{(y - x)}^{2}}$ .

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

Step 3.5.2

Reorder terms.

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

Step 4

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

$z' = \frac{y}{{(- x + y)}^{2}}$

Step 5

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

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