Algebra Examples

$\frac{x + 3}{y (z + 5)}$

Step 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 $f (x) = x + 3$ and $g (x) = y (z + 5)$ .

$\frac{y (z + 5) \frac{d}{d x} [x + 3] - (x + 3) \frac{d}{d x} [y (z + 5)]}{{(y (z + 5))}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{y (z + 5) (\frac{d}{d x} [x] + \frac{d}{d x} [3]) - (x + 3) \frac{d}{d x} [y (z + 5)]}{{(y (z + 5))}^{2}}$

Step 2.2

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

$\frac{y (z + 5) (1 + \frac{d}{d x} [3]) - (x + 3) \frac{d}{d x} [y (z + 5)]}{{(y (z + 5))}^{2}}$

Step 2.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{y (z + 5) (1 + 0) - (x + 3) \frac{d}{d x} [y (z + 5)]}{{(y (z + 5))}^{2}}$

Step 2.4

Add $1$ and $0$ .

$\frac{y (z + 5) \cdot 1 - (x + 3) \frac{d}{d x} [y (z + 5)]}{{(y (z + 5))}^{2}}$

Step 2.5

Since $y (z + 5)$ is constant with respect to $x$ , the derivative of $y (z + 5)$ with respect to $x$ is $0$ .

$\frac{y (z + 5) \cdot 1 - (x + 3) \cdot 0}{{(y (z + 5))}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $y (z + 5)$ .

$\frac{y (z + 5) \cdot 1 - (x + 3) \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{(y z + y \cdot 5) \cdot 1 - (x + 3) \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{y z \cdot 1 + y \cdot 5 \cdot 1 - (x + 3) \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.4

Apply the distributive property.

$\frac{y z \cdot 1 + y \cdot 5 \cdot 1 + (- x - 1 \cdot 3) \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.5

Apply the distributive property.

$\frac{y z \cdot 1 + y \cdot 5 \cdot 1 - x \cdot 0 - 1 \cdot 3 \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Simplify each term.

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Step 3.6.1.1

Multiply $y$ by $1$ .

$\frac{y z + y \cdot 5 \cdot 1 - x \cdot 0 - 1 \cdot 3 \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.6.1.2

Move $5$ to the left of $y$ .

$\frac{y z + 5 \cdot y \cdot 1 - x \cdot 0 - 1 \cdot 3 \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.6.1.3

Multiply $5$ by $1$ .

$\frac{y z + 5 y - x \cdot 0 - 1 \cdot 3 \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.6.1.4

Multiply $- x \cdot 0$ .

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Step 3.6.1.4.1

Multiply $0$ by $- 1$ .

$\frac{y z + 5 y + 0 x - 1 \cdot 3 \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.6.1.4.2

Multiply $0$ by $x$ .

$\frac{y z + 5 y + 0 - 1 \cdot 3 \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.6.1.5

Multiply $- 1 \cdot 3 \cdot 0$ .

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Step 3.6.1.5.1

Multiply $- 1$ by $3$ .

$\frac{y z + 5 y + 0 - 3 \cdot 0}{y^{2} {(z + 5)}^{2}}$

Step 3.6.1.5.2

Multiply $- 3$ by $0$ .

$\frac{y z + 5 y + 0 + 0}{y^{2} {(z + 5)}^{2}}$

Step 3.6.2

Combine the opposite terms in $y z + 5 y + 0 + 0$ .

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Step 3.6.2.1

Add $y z + 5 y$ and $0$ .

$\frac{y z + 5 y + 0}{y^{2} {(z + 5)}^{2}}$

Step 3.6.2.2

Add $y z + 5 y$ and $0$ .

$\frac{y z + 5 y}{y^{2} {(z + 5)}^{2}}$

Step 3.7

Factor $y$ out of $y z + 5 y$ .

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Step 3.7.1

Factor $y$ out of $y z$ .

$\frac{y (z) + 5 y}{y^{2} {(z + 5)}^{2}}$

Step 3.7.2

Factor $y$ out of $5 y$ .

$\frac{y (z) + y \cdot 5}{y^{2} {(z + 5)}^{2}}$

Step 3.7.3

Factor $y$ out of $y (z) + y \cdot 5$ .

$\frac{y (z + 5)}{y^{2} {(z + 5)}^{2}}$

Step 3.8

Cancel the common factors.

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Step 3.8.1

Factor $y$ out of $y^{2} {(z + 5)}^{2}$ .

$\frac{y (z + 5)}{y (y {(z + 5)}^{2})}$

Step 3.8.2

Cancel the common factor.

$\frac{y (z + 5)}{y (y {(z + 5)}^{2})}$

Step 3.8.3

Rewrite the expression.

$\frac{z + 5}{y {(z + 5)}^{2}}$

Step 3.9

Cancel the common factor of $z + 5$ and ${(z + 5)}^{2}$ .

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Step 3.9.1

Multiply by $1$ .

$\frac{(z + 5) \cdot 1}{y {(z + 5)}^{2}}$

Step 3.9.2

Cancel the common factors.

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Step 3.9.2.1

Factor $z + 5$ out of $y {(z + 5)}^{2}$ .

$\frac{(z + 5) \cdot 1}{(z + 5) (y (z + 5))}$

Step 3.9.2.2

Cancel the common factor.

$\frac{(z + 5) \cdot 1}{(z + 5) (y (z + 5))}$

Step 3.9.2.3

Rewrite the expression.

$\frac{1}{y (z + 5)}$