Calculus Examples

$\log_{5} (2 x - 6)$

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 $g (x) = 2 x - 6$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $2 x - 6$ .

$\frac{d}{d u} [\log_{5} (u)] \frac{d}{d x} [2 x - 6]$

Step 1.2

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

$\frac{1}{u \ln (5)} \frac{d}{d x} [2 x - 6]$

Step 1.3

Replace all occurrences of $u$ with $2 x - 6$ .

$\frac{1}{(2 x - 6) \ln (5)} \frac{d}{d x} [2 x - 6]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $2 x - 6$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6]$ .

$\frac{1}{(2 x - 6) \ln (5)} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6])$

Step 2.2

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

$\frac{1}{(2 x - 6) \ln (5)} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 6])$

Step 2.3

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

$\frac{1}{(2 x - 6) \ln (5)} (2 \cdot 1 + \frac{d}{d x} [- 6])$

Step 2.4

Multiply $2$ by $1$ .

$\frac{1}{(2 x - 6) \ln (5)} (2 + \frac{d}{d x} [- 6])$

Step 2.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{1}{(2 x - 6) \ln (5)} (2 + 0)$

Step 2.6

Simplify terms.

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Step 2.6.1

Add $2$ and $0$ .

$\frac{1}{(2 x - 6) \ln (5)} \cdot 2$

Step 2.6.2

Combine $\frac{1}{(2 x - 6) \ln (5)}$ and $2$ .

$\frac{2}{(2 x - 6) \ln (5)}$

Step 2.6.3

Cancel the common factor of $2$ and $2 x - 6$ .

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Step 2.6.3.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 1}{(2 x - 6) \ln (5)}$

Step 2.6.3.2

Cancel the common factors.

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Step 2.6.3.2.1

Factor $2$ out of $(2 x - 6) \ln (5)$ .

$\frac{2 \cdot 1}{2 ((x - 3) \ln (5))}$

Step 2.6.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 ((x - 3) \ln (5))}$

Step 2.6.3.2.3

Rewrite the expression.

$\frac{1}{(x - 3) \ln (5)}$

Step 2.6.4

Apply the distributive property.

$\frac{1}{x \ln (5) - 3 \ln (5)}$