Calculus Examples

$y = 5 x^{2} \ln (\frac{x}{4})$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2} \ln (\frac{x}{4})$ with respect to $x$ is $5 \frac{d}{d x} [x^{2} \ln (\frac{x}{4})]$ .

$5 \frac{d}{d x} [x^{2} \ln (\frac{x}{4})]$

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 $g (x) = \ln (\frac{x}{4})$ .

$5 (x^{2} \frac{d}{d x} [\ln (\frac{x}{4})] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

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 $g (x) = \frac{x}{4}$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u$ as $\frac{x}{4}$ .

$5 (x^{2} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{x}{4}]) + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 3.2

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

$5 (x^{2} (\frac{1}{u} \frac{d}{d x} [\frac{x}{4}]) + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 3.3

Replace all occurrences of $u$ with $\frac{x}{4}$ .

$5 (x^{2} (\frac{1}{\frac{x}{4}} \frac{d}{d x} [\frac{x}{4}]) + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 4

Multiply by the reciprocal of the fraction to divide by $\frac{x}{4}$ .

$5 (x^{2} (1 \frac{4}{x} \frac{d}{d x} [\frac{x}{4}]) + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Multiply $\frac{4}{x}$ by $1$ .

$5 (x^{2} (\frac{4}{x} \frac{d}{d x} [\frac{x}{4}]) + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 5.2

Combine $\frac{4}{x}$ and $x^{2}$ .

$5 (\frac{4 x^{2}}{x} \frac{d}{d x} [\frac{x}{4}] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Factor $x$ out of $4 x^{2}$ .

$5 (\frac{x (4 x)}{x} \frac{d}{d x} [\frac{x}{4}] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 5.3.2

Cancel the common factors.

Tap for more steps...

Step 5.3.2.1

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

$5 (\frac{x (4 x)}{x^{1}} \frac{d}{d x} [\frac{x}{4}] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 5.3.2.2

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

$5 (\frac{x (4 x)}{x \cdot 1} \frac{d}{d x} [\frac{x}{4}] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 5.3.2.3

Cancel the common factor.

$5 (\frac{x (4 x)}{x \cdot 1} \frac{d}{d x} [\frac{x}{4}] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 5.3.2.4

Rewrite the expression.

$5 (\frac{4 x}{1} \frac{d}{d x} [\frac{x}{4}] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 5.3.2.5

Divide $4 x$ by $1$ .

$5 (4 x \frac{d}{d x} [\frac{x}{4}] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 6

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x]$ .

$5 (4 x (\frac{1}{4} \frac{d}{d x} [x]) + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 7

Simplify terms.

Tap for more steps...

Step 7.1

Combine $4$ and $\frac{1}{4}$ .

$5 (x (\frac{4}{4} \frac{d}{d x} [x]) + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 7.2

Combine $\frac{4}{4}$ and $x$ .

$5 (\frac{4 x}{4} \frac{d}{d x} [x] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 7.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 7.3.1

Cancel the common factor.

$5 (\frac{4 x}{4} \frac{d}{d x} [x] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 7.3.2

Divide $x$ by $1$ .

$5 (x \frac{d}{d x} [x] + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 8

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

$5 (x \cdot 1 + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 9

Multiply $x$ by $1$ .

$5 (x + \ln (\frac{x}{4}) \frac{d}{d x} [x^{2}])$

Step 10

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

$5 (x + \ln (\frac{x}{4}) (2 x))$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Apply the distributive property.

$5 x + 5 (\ln (\frac{x}{4}) (2 x))$

Step 11.2

Multiply $2$ by $5$ .

$5 x + 10 \ln (\frac{x}{4}) x$

Step 11.3

Reorder terms.

$10 x \ln (\frac{x}{4}) + 5 x$