Calculus Examples

$f (x) = x (1 - \frac{4}{x + 3})$

Step 1

Simplify terms.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$\frac{d}{d x} [x (\frac{x + 3}{x + 3} - \frac{4}{x + 3})]$

Step 1.2

Combine the numerators over the common denominator.

$\frac{d}{d x} [x \frac{x + 3 - 4}{x + 3}]$

Step 1.3

Subtract $4$ from $3$ .

$\frac{d}{d x} [x \frac{x - 1}{x + 3}]$

Step 1.4

Combine $x$ and $\frac{x - 1}{x + 3}$ .

$\frac{d}{d x} [\frac{x (x - 1)}{x + 3}]$

Step 2

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 $f (x) = x (x - 1)$ and $g (x) = x + 3$ .

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

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

Step 4.4.2

Multiply $x$ by $1$ .

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

Step 4.5

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

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

Step 4.6

Simplify by adding terms.

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Step 4.6.1

Multiply $x - 1$ by $1$ .

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

Step 4.6.2

Add $x$ and $x$ .

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

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

$\frac{(x + 3) (2 x - 1) - x (x - 1) (1 + 0)}{{(x + 3)}^{2}}$

Step 4.10

Simplify the expression.

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Step 4.10.1

Add $1$ and $0$ .

$\frac{(x + 3) (2 x - 1) - x (x - 1) \cdot 1}{{(x + 3)}^{2}}$

Step 4.10.2

Multiply $- 1$ by $1$ .

$\frac{(x + 3) (2 x - 1) - x (x - 1)}{{(x + 3)}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{(x + 3) (2 x - 1) - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Expand $(x + 3) (2 x - 1)$ using the FOIL Method.

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Step 5.2.1.1.1

Apply the distributive property.

$\frac{x (2 x - 1) + 3 (2 x - 1) - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2.1.1.2

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 1 + 3 (2 x - 1) - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2.1.1.3

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 1 + 3 (2 x) + 3 \cdot - 1 - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2.1.2

Simplify and combine like terms.

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Step 5.2.1.2.1

Simplify each term.

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Step 5.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x \cdot - 1 + 3 (2 x) + 3 \cdot - 1 - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2.1.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.1.2.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot - 1 + 3 (2 x) + 3 \cdot - 1 - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot - 1 + 3 (2 x) + 3 \cdot - 1 - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2.1.2.1.3

Move $- 1$ to the left of $x$ .

$\frac{2 x^{2} - 1 \cdot x + 3 (2 x) + 3 \cdot - 1 - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2.1.2.1.4

Rewrite $- 1 x$ as $- x$ .

$\frac{2 x^{2} - x + 3 (2 x) + 3 \cdot - 1 - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2.1.2.1.5

Multiply $2$ by $3$ .

$\frac{2 x^{2} - x + 6 x + 3 \cdot - 1 - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2.1.2.1.6

Multiply $3$ by $- 1$ .

$\frac{2 x^{2} - x + 6 x - 3 - x \cdot x - x \cdot - 1}{{(x + 3)}^{2}}$

Step 5.2.1.2.2

Add $- x$ and $6 x$ .

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

Step 5.2.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.1.3.1

Move $x$ .

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

Step 5.2.1.3.2

Multiply $x$ by $x$ .

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

Step 5.2.1.4

Multiply $- x \cdot - 1$ .

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Step 5.2.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{2} + 5 x - 3 - x^{2} + 1 x}{{(x + 3)}^{2}}$

Step 5.2.1.4.2

Multiply $x$ by $1$ .

$\frac{2 x^{2} + 5 x - 3 - x^{2} + x}{{(x + 3)}^{2}}$

Step 5.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{x^{2} + 5 x - 3 + x}{{(x + 3)}^{2}}$

Step 5.2.3

Add $5 x$ and $x$ .

$\frac{x^{2} + 6 x - 3}{{(x + 3)}^{2}}$