Calculus Examples

$f (x) = 5 x^{5} \ln (x)$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{5} \ln (x)$ with respect to $x$ is $5 \frac{d}{d x} [x^{5} \ln (x)]$ .

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

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 $f (x) = x^{5}$ and $g (x) = \ln (x)$ .

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

Combine $x^{5}$ and $\frac{1}{x}$ .

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

Step 4.2

Cancel the common factor of $x^{5}$ and $x$ .

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Step 4.2.1

Factor $x$ out of $x^{5}$ .

$5 (\frac{x \cdot x^{4}}{x} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Raise $x$ to the power of $1$ .

$5 (\frac{x \cdot x^{4}}{x^{1}} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2.2.2

Factor $x$ out of $x^{1}$ .

$5 (\frac{x \cdot x^{4}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2.2.3

Cancel the common factor.

$5 (\frac{x \cdot x^{4}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2.2.4

Rewrite the expression.

$5 (\frac{x^{4}}{1} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.2.2.5

Divide $x^{4}$ by $1$ .

$5 (x^{4} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 (x^{4} + \ln (x) (5 x^{4}))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$5 x^{4} + 5 (\ln (x) (5 x^{4}))$

Step 5.2

Multiply $5$ by $5$ .

$5 x^{4} + 25 \ln (x) x^{4}$

Step 5.3

Reorder terms.

$5 x^{4} + 25 x^{4} \ln (x)$