Calculus Examples

$\log_{5} (x^{8} + x)$

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 $g (x) = x^{8} + x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{8} + x$ .

$\frac{d}{d u} [\log_{5} (u)] \frac{d}{d x} [x^{8} + x]$

Step 1.2

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

$\frac{1}{u \ln (5)} \frac{d}{d x} [x^{8} + x]$

Step 1.3

Replace all occurrences of $u$ with $x^{8} + x$ .

$\frac{1}{(x^{8} + x) \ln (5)} \frac{d}{d x} [x^{8} + x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{8} + x$ with respect to $x$ is $\frac{d}{d x} [x^{8}] + \frac{d}{d x} [x]$ .

$\frac{1}{(x^{8} + x) \ln (5)} (\frac{d}{d x} [x^{8}] + \frac{d}{d x} [x])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$\frac{1}{(x^{8} + x) \ln (5)} (8 x^{7} + \frac{d}{d x} [x])$

Step 2.3

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

$\frac{1}{(x^{8} + x) \ln (5)} (8 x^{7} + 1)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{1}{x^{8} \ln (5) + x \ln (5)} (8 x^{7} + 1)$

Step 3.2

Reorder the factors of $\frac{1}{x^{8} \ln (5) + x \ln (5)} (8 x^{7} + 1)$ .

$(8 x^{7} + 1) \frac{1}{x^{8} \ln (5) + x \ln (5)}$

Step 3.3

Factor $x \ln (5)$ out of $x^{8} \ln (5) + x \ln (5)$ .

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Step 3.3.1

Factor $x \ln (5)$ out of $x^{8} \ln (5)$ .

$(8 x^{7} + 1) \frac{1}{x \ln (5) (x^{7}) + x \ln (5)}$

Step 3.3.2

Factor $x \ln (5)$ out of $x \ln (5)$ .

$(8 x^{7} + 1) \frac{1}{x \ln (5) (x^{7}) + x \ln (5) (1)}$

Step 3.3.3

Factor $x \ln (5)$ out of $x \ln (5) (x^{7}) + x \ln (5) (1)$ .

$(8 x^{7} + 1) \frac{1}{x \ln (5) (x^{7} + 1)}$

Step 3.4

Multiply $8 x^{7} + 1$ by $\frac{1}{x \ln (5) (x^{7} + 1)}$ .

$\frac{8 x^{7} + 1}{x \ln (5) (x^{7} + 1)}$