Calculus Examples

$N (x) = 340 + 84 \ln (2 + 3.5 x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $340 + 84 \ln (2 + 3.5 x)$ with respect to $x$ is $\frac{d}{d x} [340] + \frac{d}{d x} [84 \ln (2 + 3.5 x)]$ .

$\frac{d}{d x} [340] + \frac{d}{d x} [84 \ln (2 + 3.5 x)]$

Step 1.2

Since $340$ is constant with respect to $x$ , the derivative of $340$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [84 \ln (2 + 3.5 x)]$

Step 2

Evaluate $\frac{d}{d x} [84 \ln (2 + 3.5 x)]$ .

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Step 2.1

Since $84$ is constant with respect to $x$ , the derivative of $84 \ln (2 + 3.5 x)$ with respect to $x$ is $84 \frac{d}{d x} [\ln (2 + 3.5 x)]$ .

$0 + 84 \frac{d}{d x} [\ln (2 + 3.5 x)]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 2 + 3.5 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $2 + 3.5 x$ .

$0 + 84 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [2 + 3.5 x])$

Step 2.2.2

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

$0 + 84 (\frac{1}{u} \frac{d}{d x} [2 + 3.5 x])$

Step 2.2.3

Replace all occurrences of $u$ with $2 + 3.5 x$ .

$0 + 84 (\frac{1}{2 + 3.5 x} \frac{d}{d x} [2 + 3.5 x])$

Step 2.3

By the Sum Rule, the derivative of $2 + 3.5 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [3.5 x]$ .

$0 + 84 (\frac{1}{2 + 3.5 x} (\frac{d}{d x} [2] + \frac{d}{d x} [3.5 x]))$

Step 2.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + 84 (\frac{1}{2 + 3.5 x} (0 + \frac{d}{d x} [3.5 x]))$

Step 2.5

Since $3.5$ is constant with respect to $x$ , the derivative of $3.5 x$ with respect to $x$ is $3.5 \frac{d}{d x} [x]$ .

$0 + 84 (\frac{1}{2 + 3.5 x} (0 + 3.5 \frac{d}{d x} [x]))$

Step 2.6

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

$0 + 84 (\frac{1}{2 + 3.5 x} (0 + 3.5 \cdot 1))$

Step 2.7

Multiply $3.5$ by $1$ .

$0 + 84 (\frac{1}{2 + 3.5 x} (0 + 3.5))$

Step 2.8

Add $0$ and $3.5$ .

$0 + 84 (\frac{1}{2 + 3.5 x} \cdot 3.5)$

Step 2.9

Combine $\frac{1}{2 + 3.5 x}$ and $3.5$ .

$0 + 84 \frac{3.5}{2 + 3.5 x}$

Step 2.10

Combine $84$ and $\frac{3.5}{2 + 3.5 x}$ .

$0 + \frac{84 \cdot 3.5}{2 + 3.5 x}$

Step 2.11

Multiply $84$ by $3.5$ .

$0 + \frac{294}{2 + 3.5 x}$

Step 3

Simplify.

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Step 3.1

Add $0$ and $\frac{294}{2 + 3.5 x}$ .

$\frac{294}{2 + 3.5 x}$

Step 3.2

Reorder terms.

$\frac{294}{3.5 x + 2}$

Step 3.3

Factor $0.5$ out of $3.5 x + 2$ .

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Step 3.3.1

Factor $0.5$ out of $3.5 x$ .

$\frac{294}{0.5 (7 x) + 2}$

Step 3.3.2

Factor $0.5$ out of $2$ .

$\frac{294}{0.5 (7 x) + 0.5 (4)}$

Step 3.3.3

Factor $0.5$ out of $0.5 (7 x) + 0.5 (4)$ .

$\frac{294}{0.5 (7 x + 4)}$

Step 3.4

Factor $294$ out of $294$ .

$\frac{294 (1)}{0.5 (7 x + 4)}$

Step 3.5

Separate fractions.

$\frac{294}{0.5} \cdot \frac{1}{7 x + 4}$

Step 3.6

Divide $294$ by $0.5$ .

$588 \frac{1}{7 x + 4}$

Step 3.7

Combine $588$ and $\frac{1}{7 x + 4}$ .

$\frac{588}{7 x + 4}$