Calculus Examples

$f (x) = 8 x^{2} \ln (8 x)$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2} \ln (8 x)$ with respect to $x$ is $8 \frac{d}{d x} [x^{2} \ln (8 x)]$ .

$8 \frac{d}{d x} [x^{2} \ln (8 x)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = \ln (8 x)$ .

$8 (x^{2} \frac{d}{d x} [\ln (8 x)] + \ln (8 x) \frac{d}{d x} [x^{2}])$

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 $g (x) = 8 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $8 x$ .

$8 (x^{2} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [8 x]) + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 3.2

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

$8 (x^{2} (\frac{1}{u} \frac{d}{d x} [8 x]) + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 3.3

Replace all occurrences of $u$ with $8 x$ .

$8 (x^{2} (\frac{1}{8 x} \frac{d}{d x} [8 x]) + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4

Differentiate.

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Step 4.1

Combine $\frac{1}{8 x}$ and $x^{2}$ .

$8 (\frac{x^{2}}{8 x} \frac{d}{d x} [8 x] + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 4.2.1

Factor $x$ out of $x^{2}$ .

$8 (\frac{x \cdot x}{8 x} \frac{d}{d x} [8 x] + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $x$ out of $8 x$ .

$8 (\frac{x \cdot x}{x \cdot 8} \frac{d}{d x} [8 x] + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.2.2.2

Cancel the common factor.

$8 (\frac{x \cdot x}{x \cdot 8} \frac{d}{d x} [8 x] + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.2.2.3

Rewrite the expression.

$8 (\frac{x}{8} \frac{d}{d x} [8 x] + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.3

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 (\frac{x}{8} (8 \frac{d}{d x} [x]) + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.4

Simplify terms.

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Step 4.4.1

Combine $8$ and $\frac{x}{8}$ .

$8 (\frac{8 x}{8} \frac{d}{d x} [x] + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.4.2

Cancel the common factor of $8$ .

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Step 4.4.2.1

Cancel the common factor.

$8 (\frac{8 x}{8} \frac{d}{d x} [x] + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.4.2.2

Divide $x$ by $1$ .

$8 (x \frac{d}{d x} [x] + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.5

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

$8 (x \cdot 1 + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.6

Multiply $x$ by $1$ .

$8 (x + \ln (8 x) \frac{d}{d x} [x^{2}])$

Step 4.7

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

$8 (x + \ln (8 x) (2 x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$8 x + 8 (\ln (8 x) (2 x))$

Step 5.2

Multiply $2$ by $8$ .

$8 x + 16 \ln (8 x) x$

Step 5.3

Reorder terms.

$16 x \ln (8 x) + 8 x$