Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $y + z$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $y + z$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$(x - z) (- {(y + z)}^{- 2} (1 + 0))$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $- 1$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Reorder the factors of $(x - z) (- \frac{1}{{(y + z)}^{2}})$ .

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Multiply $- - z$ .

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.4.2

Multiply $z$ by $1$ .

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

Step 4.5

Multiply $- x + z$ by $\frac{1}{{(y + z)}^{2}}$ .

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

Step 4.6

Factor $- 1$ out of $- x$ .

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

Step 4.7

Factor $- 1$ out of $z$ .

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

Step 4.8

Factor $- 1$ out of $- (x) - 1 (- z)$ .

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

Step 4.9

Rewrite $- (x - z)$ as $- 1 (x - z)$ .

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

Step 4.10

Move the negative in front of the fraction.

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