Calculus Examples

$f (x) = 4 - x^{2} - \ln (\frac{1}{2} x + 1)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $4 - x^{2} - \ln (\frac{1}{2} x + 1)$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- \ln (\frac{1}{2} x + 1)]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- \ln (\frac{1}{2} x + 1)]$

Step 1.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 2

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2.3

Multiply $2$ by $- 1$ .

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

Step 3

Evaluate $\frac{d}{d x} [- \ln (\frac{1}{2} x + 1)]$ .

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Step 3.1

Combine $\frac{1}{2}$ and $x$ .

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

Step 3.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (\frac{x}{2} + 1)$ with respect to $x$ is $- \frac{d}{d x} [\ln (\frac{x}{2} + 1)]$ .

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

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{x}{2} + 1$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $\frac{x}{2} + 1$ .

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

Step 3.3.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 3.3.3

Replace all occurrences of $u$ with $\frac{x}{2} + 1$ .

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

Step 3.4

By the Sum Rule, the derivative of $\frac{x}{2} + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{x}{2}] + \frac{d}{d x} [1]$ .

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

Step 3.5

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Multiply $\frac{1}{2}$ by $1$ .

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

Step 3.9

Add $\frac{1}{2}$ and $0$ .

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

Step 3.10

Multiply $\frac{1}{\frac{x}{2} + 1}$ by $\frac{1}{2}$ .

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

Step 3.11

Move $2$ to the left of $\frac{x}{2} + 1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine terms.

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Step 4.2.1

Subtract $2 x$ from $0$ .

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

Step 4.2.2

Combine $2$ and $\frac{x}{2}$ .

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

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Cancel the common factor.

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

Step 4.2.3.2

Divide $x$ by $1$ .

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

Step 4.2.4

Multiply $2$ by $1$ .

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

Step 4.2.5

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 4.2.6

Combine $- 2 x$ and $\frac{x + 2}{x + 2}$ .

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

Step 4.2.7

Combine the numerators over the common denominator.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Apply the distributive property.

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

Step 4.3.2

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

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Step 4.3.2.1

Move $x$ .

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

Step 4.3.2.2

Multiply $x$ by $x$ .

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

Step 4.3.3

Multiply $2$ by $- 2$ .

$\frac{- 2 x^{2} - 4 x - 1}{x + 2}$

Step 4.4

Factor $- 1$ out of $- 2 x^{2}$ .

$\frac{- (2 x^{2}) - 4 x - 1}{x + 2}$

Step 4.5

Factor $- 1$ out of $- 4 x$ .

$\frac{- (2 x^{2}) - (4 x) - 1}{x + 2}$

Step 4.6

Factor $- 1$ out of $- (2 x^{2}) - (4 x)$ .

$\frac{- (2 x^{2} + 4 x) - 1}{x + 2}$

Step 4.7

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

$\frac{- (2 x^{2} + 4 x) - 1 (1)}{x + 2}$

Step 4.8

Factor $- 1$ out of $- (2 x^{2} + 4 x) - 1 (1)$ .

$\frac{- (2 x^{2} + 4 x + 1)}{x + 2}$

Step 4.9

Rewrite $- (2 x^{2} + 4 x + 1)$ as $- 1 (2 x^{2} + 4 x + 1)$ .

$\frac{- 1 (2 x^{2} + 4 x + 1)}{x + 2}$

Step 4.10

Move the negative in front of the fraction.

$- \frac{2 x^{2} + 4 x + 1}{x + 2}$