Calculus Examples

$f (x) = 3 x^{4} \ln ({(x)}^{2})$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4} \ln (x^{2})$ with respect to $x$ is $3 \frac{d}{d x} [x^{4} \ln (x^{2})]$ .

$3 \frac{d}{d x} [x^{4} \ln (x^{2})]$

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 $f (x) = x^{4}$ and $g (x) = \ln (x^{2})$ .

$3 (x^{4} \frac{d}{d x} [\ln (x^{2})] + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

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 $g (x) = x^{2}$ .

Tap for more steps...

Step 3.1

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

$3 (x^{4} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2}]) + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 3.2

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

$3 (x^{4} (\frac{1}{u} \frac{d}{d x} [x^{2}]) + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 3.3

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

$3 (x^{4} (\frac{1}{x^{2}} \frac{d}{d x} [x^{2}]) + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 4

Differentiate using the Power Rule.

Tap for more steps...

Step 4.1

Combine $\frac{1}{x^{2}}$ and $x^{4}$ .

$3 (\frac{x^{4}}{x^{2}} \frac{d}{d x} [x^{2}] + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 4.2

Cancel the common factor of $x^{4}$ and $x^{2}$ .

Tap for more steps...

Step 4.2.1

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

$3 (\frac{x^{2} x^{2}}{x^{2}} \frac{d}{d x} [x^{2}] + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 4.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.1

Multiply by $1$ .

$3 (\frac{x^{2} x^{2}}{x^{2} \cdot 1} \frac{d}{d x} [x^{2}] + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 4.2.2.2

Cancel the common factor.

$3 (\frac{x^{2} x^{2}}{x^{2} \cdot 1} \frac{d}{d x} [x^{2}] + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 4.2.2.3

Rewrite the expression.

$3 (\frac{x^{2}}{1} \frac{d}{d x} [x^{2}] + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 4.2.2.4

Divide $x^{2}$ by $1$ .

$3 (x^{2} \frac{d}{d x} [x^{2}] + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 4.3

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

$3 (x^{2} (2 x) + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 5

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 5.1

Move $x$ .

$3 (x \cdot x^{2} \cdot 2 + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 5.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 5.2.1

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

$3 (x^{1} x^{2} \cdot 2 + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 (x^{1 + 2} \cdot 2 + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 5.3

Add $1$ and $2$ .

$3 (x^{3} \cdot 2 + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 6

Move $2$ to the left of $x^{3}$ .

$3 (2 \cdot x^{3} + \ln (x^{2}) \frac{d}{d x} [x^{4}])$

Step 7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$3 (2 x^{3} + \ln (x^{2}) (4 x^{3}))$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Apply the distributive property.

$3 (2 x^{3}) + 3 (\ln (x^{2}) (4 x^{3}))$

Step 8.2

Combine terms.

Tap for more steps...

Step 8.2.1

Multiply $2$ by $3$ .

$6 x^{3} + 3 (\ln (x^{2}) (4 x^{3}))$

Step 8.2.2

Multiply $4$ by $3$ .

$6 x^{3} + 12 \ln (x^{2}) x^{3}$

Step 8.3

Reorder terms.

$6 x^{3} + 12 x^{3} \ln (x^{2})$