Calculus Examples

$\log_{5} (1 + 2 x)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{5} (x)$ and $g (x) = 1 + 2 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $1 + 2 x$ .

$\frac{d}{d u} [\log_{5} (u)] \frac{d}{d x} [1 + 2 x]$

Step 1.2

The derivative of $\log_{5} (u)$ with respect to $u$ is $\frac{1}{u \ln (5)}$ .

$\frac{1}{u \ln (5)} \frac{d}{d x} [1 + 2 x]$

Step 1.3

Replace all occurrences of $u$ with $1 + 2 x$ .

$\frac{1}{(1 + 2 x) \ln (5)} \frac{d}{d x} [1 + 2 x]$

Step 2

Differentiate.

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Step 2.1

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

$\frac{1}{(1 + 2 x) \ln (5)} (\frac{d}{d x} [1] + \frac{d}{d x} [2 x])$

Step 2.2

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

$\frac{1}{(1 + 2 x) \ln (5)} (0 + \frac{d}{d x} [2 x])$

Step 2.3

Add $0$ and $\frac{d}{d x} [2 x]$ .

$\frac{1}{(1 + 2 x) \ln (5)} \frac{d}{d x} [2 x]$

Step 2.4

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

$\frac{1}{(1 + 2 x) \ln (5)} (2 \frac{d}{d x} [x])$

Step 2.5

Combine $2$ and $\frac{1}{(1 + 2 x) \ln (5)}$ .

$\frac{2}{(1 + 2 x) \ln (5)} \frac{d}{d x} [x]$

Step 2.6

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

$\frac{2}{(1 + 2 x) \ln (5)} \cdot 1$

Step 2.7

Multiply $\frac{2}{(1 + 2 x) \ln (5)}$ by $1$ .

$\frac{2}{(1 + 2 x) \ln (5)}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{2}{1 \ln (5) + 2 x \ln (5)}$

Step 3.2

Multiply $\ln (5)$ by $1$ .

$\frac{2}{\ln (5) + 2 x \ln (5)}$

Step 3.3

Reorder terms.

$\frac{2}{2 x \ln (5) + \ln (5)}$