Calculus Examples

$\log_{a} (2 x^{3} + 7)$

Step 1

Rewrite $\log_{a} (2 x^{3} + 7)$ using the change of base formula.

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Step 1.1

The change of base rule can be used if $a$ and $b$ are greater than 0 and not equal to 1, and $x$ is greater than 0.

$\frac{d}{d x} [\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}]$

Step 1.2

Substitute in values for the variables in the change of base formula, using $b = e$ .

$\frac{d}{d x} [\frac{\ln (2 x^{3} + 7)}{\ln (a)}]$

Step 2

Since $\frac{1}{\ln (a)}$ is constant with respect to $x$ , the derivative of $\frac{\ln (2 x^{3} + 7)}{\ln (a)}$ with respect to $x$ is $\frac{1}{\ln (a)} \frac{d}{d x} [\ln (2 x^{3} + 7)]$ .

$\frac{1}{\ln (a)} \frac{d}{d x} [\ln (2 x^{3} + 7)]$

Step 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 $g (x) = 2 x^{3} + 7$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $2 x^{3} + 7$ .

$\frac{1}{\ln (a)} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [2 x^{3} + 7])$

Step 3.2

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

$\frac{1}{\ln (a)} (\frac{1}{u} \frac{d}{d x} [2 x^{3} + 7])$

Step 3.3

Replace all occurrences of $u$ with $2 x^{3} + 7$ .

$\frac{1}{\ln (a)} (\frac{1}{2 x^{3} + 7} \frac{d}{d x} [2 x^{3} + 7])$

Step 4

Differentiate.

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Step 4.1

Multiply $\frac{1}{2 x^{3} + 7}$ by $\frac{1}{\ln (a)}$ .

$\frac{1}{(2 x^{3} + 7) \ln (a)} \frac{d}{d x} [2 x^{3} + 7]$

Step 4.2

By the Sum Rule, the derivative of $2 x^{3} + 7$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [7]$ .

$\frac{1}{(2 x^{3} + 7) \ln (a)} (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [7])$

Step 4.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{(2 x^{3} + 7) \ln (a)} (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [7])$

Step 4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{(2 x^{3} + 7) \ln (a)} (2 (3 x^{2}) + \frac{d}{d x} [7])$

Step 4.5

Multiply $3$ by $2$ .

$\frac{1}{(2 x^{3} + 7) \ln (a)} (6 x^{2} + \frac{d}{d x} [7])$

Step 4.6

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{1}{(2 x^{3} + 7) \ln (a)} (6 x^{2} + 0)$

Step 4.7

Combine fractions.

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Step 4.7.1

Add $6 x^{2}$ and $0$ .

$\frac{1}{(2 x^{3} + 7) \ln (a)} (6 x^{2})$

Step 4.7.2

Combine $6$ and $\frac{1}{(2 x^{3} + 7) \ln (a)}$ .

$\frac{6}{(2 x^{3} + 7) \ln (a)} x^{2}$

Step 4.7.3

Combine $\frac{6}{(2 x^{3} + 7) \ln (a)}$ and $x^{2}$ .

$\frac{6 x^{2}}{(2 x^{3} + 7) \ln (a)}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{6 x^{2}}{2 x^{3} \ln (a) + 7 \ln (a)}$

Step 5.2

Factor $\ln (a)$ out of $2 x^{3} \ln (a) + 7 \ln (a)$ .

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Step 5.2.1

Factor $\ln (a)$ out of $2 x^{3} \ln (a)$ .

$\frac{6 x^{2}}{\ln (a) (2 x^{3}) + 7 \ln (a)}$

Step 5.2.2

Factor $\ln (a)$ out of $7 \ln (a)$ .

$\frac{6 x^{2}}{\ln (a) (2 x^{3}) + \ln (a) \cdot 7}$

Step 5.2.3

Factor $\ln (a)$ out of $\ln (a) (2 x^{3}) + \ln (a) \cdot 7$ .

$\frac{6 x^{2}}{\ln (a) (2 x^{3} + 7)}$