Calculus Examples

$\log_{2} (x^{2} + 10 x)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{2} (x)$ and $g (x) = x^{2} + 10 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} + 10 x$ .

$\frac{d}{d u} [\log_{2} (u)] \frac{d}{d x} [x^{2} + 10 x]$

Step 1.2

The derivative of $\log_{2} (u)$ with respect to $u$ is $\frac{1}{u \ln (2)}$ .

$\frac{1}{u \ln (2)} \frac{d}{d x} [x^{2} + 10 x]$

Step 1.3

Replace all occurrences of $u$ with $x^{2} + 10 x$ .

$\frac{1}{(x^{2} + 10 x) \ln (2)} \frac{d}{d x} [x^{2} + 10 x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} + 10 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x]$ .

$\frac{1}{(x^{2} + 10 x) \ln (2)} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{(x^{2} + 10 x) \ln (2)} (2 x + \frac{d}{d x} [10 x])$

Step 2.3

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$\frac{1}{(x^{2} + 10 x) \ln (2)} (2 x + 10 \frac{d}{d x} [x])$

Step 2.4

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

$\frac{1}{(x^{2} + 10 x) \ln (2)} (2 x + 10 \cdot 1)$

Step 2.5

Multiply $10$ by $1$ .

$\frac{1}{(x^{2} + 10 x) \ln (2)} (2 x + 10)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{1}{x^{2} \ln (2) + 10 x \ln (2)} (2 x + 10)$

Step 3.2

Reorder the factors of $\frac{1}{x^{2} \ln (2) + 10 x \ln (2)} (2 x + 10)$ .

$(2 x + 10) \frac{1}{x^{2} \ln (2) + 10 x \ln (2)}$

Step 3.3

Factor $x \ln (2)$ out of $x^{2} \ln (2) + 10 x \ln (2)$ .

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Step 3.3.1

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

$(2 x + 10) \frac{1}{x \ln (2) (x) + 10 x \ln (2)}$

Step 3.3.2

Factor $x \ln (2)$ out of $10 x \ln (2)$ .

$(2 x + 10) \frac{1}{x \ln (2) (x) + x \ln (2) (10)}$

Step 3.3.3

Factor $x \ln (2)$ out of $x \ln (2) (x) + x \ln (2) (10)$ .

$(2 x + 10) \frac{1}{x \ln (2) (x + 10)}$

Step 3.4

Multiply $2 x + 10$ by $\frac{1}{x \ln (2) (x + 10)}$ .

$\frac{2 x + 10}{x \ln (2) (x + 10)}$

Step 3.5

Factor $2$ out of $2 x + 10$ .

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Step 3.5.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 10}{x \ln (2) (x + 10)}$

Step 3.5.2

Factor $2$ out of $10$ .

$\frac{2 x + 2 \cdot 5}{x \ln (2) (x + 10)}$

Step 3.5.3

Factor $2$ out of $2 x + 2 \cdot 5$ .

$\frac{2 (x + 5)}{x \ln (2) (x + 10)}$