Calculus Examples

$\frac{46 q}{\ln (q)} - 94$

Step 1

By the Sum Rule, the derivative of $\frac{46 q}{\ln (q)} - 94$ with respect to $q$ is $\frac{d}{d q} [\frac{46 q}{\ln (q)}] + \frac{d}{d q} [- 94]$ .

$\frac{d}{d q} [\frac{46 q}{\ln (q)}] + \frac{d}{d q} [- 94]$

Step 2

Evaluate $\frac{d}{d q} [\frac{46 q}{\ln (q)}]$ .

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Step 2.1

Since $46$ is constant with respect to $q$ , the derivative of $\frac{46 q}{\ln (q)}$ with respect to $q$ is $46 \frac{d}{d q} [\frac{q}{\ln (q)}]$ .

$46 \frac{d}{d q} [\frac{q}{\ln (q)}] + \frac{d}{d q} [- 94]$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d q} [\frac{f (q)}{g (q)}]$ is $\frac{g (q) \frac{d}{d q} [f (q)] - f (q) \frac{d}{d q} [g (q)]}{{g (q)}^{2}}$ where $f (q) = q$ and $g (q) = \ln (q)$ .

$46 \frac{\ln (q) \frac{d}{d q} [q] - q \frac{d}{d q} [\ln (q)]}{\ln^{2} (q)} + \frac{d}{d q} [- 94]$

Step 2.3

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

$46 \frac{\ln (q) \cdot 1 - q \frac{d}{d q} [\ln (q)]}{\ln^{2} (q)} + \frac{d}{d q} [- 94]$

Step 2.4

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

$46 \frac{\ln (q) \cdot 1 - q \frac{1}{q}}{\ln^{2} (q)} + \frac{d}{d q} [- 94]$

Step 2.5

Multiply $\ln (q)$ by $1$ .

$46 \frac{\ln (q) - q \frac{1}{q}}{\ln^{2} (q)} + \frac{d}{d q} [- 94]$

Step 2.6

Combine $\frac{1}{q}$ and $q$ .

$46 \frac{\ln (q) - \frac{q}{q}}{\ln^{2} (q)} + \frac{d}{d q} [- 94]$

Step 2.7

Cancel the common factor of $q$ .

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Step 2.7.1

Cancel the common factor.

$46 \frac{\ln (q) - \frac{q}{q}}{\ln^{2} (q)} + \frac{d}{d q} [- 94]$

Step 2.7.2

Rewrite the expression.

$46 \frac{\ln (q) - 1 \cdot 1}{\ln^{2} (q)} + \frac{d}{d q} [- 94]$

Step 2.8

Multiply $- 1$ by $1$ .

$46 \frac{\ln (q) - 1}{\ln^{2} (q)} + \frac{d}{d q} [- 94]$

Step 2.9

Combine $46$ and $\frac{\ln (q) - 1}{\ln^{2} (q)}$ .

$\frac{46 (\ln (q) - 1)}{\ln^{2} (q)} + \frac{d}{d q} [- 94]$

Step 3

Since $- 94$ is constant with respect to $q$ , the derivative of $- 94$ with respect to $q$ is $0$ .

$\frac{46 (\ln (q) - 1)}{\ln^{2} (q)} + 0$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{46 \ln (q) + 46 \cdot - 1}{\ln^{2} (q)} + 0$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $46$ by $- 1$ .

$\frac{46 \ln (q) - 46}{\ln^{2} (q)} + 0$

Step 4.2.2

Add $\frac{46 \ln (q) - 46}{\ln^{2} (q)}$ and $0$ .

$\frac{46 \ln (q) - 46}{\ln^{2} (q)}$

Step 4.3

Factor $46$ out of $46 \ln (q) - 46$ .

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Step 4.3.1

Factor $46$ out of $46 \ln (q)$ .

$\frac{46 (\ln (q)) - 46}{\ln^{2} (q)}$

Step 4.3.2

Factor $46$ out of $- 46$ .

$\frac{46 (\ln (q)) + 46 (- 1)}{\ln^{2} (q)}$

Step 4.3.3

Factor $46$ out of $46 (\ln (q)) + 46 (- 1)$ .

$\frac{46 (\ln (q) - 1)}{\ln^{2} (q)}$