Calculus Examples

$y = \ln (4 x^{2}) \cdot (- x^{3} - 4)$

Step 1

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 $f (x) = \ln (4 x^{2})$ and $g (x) = - x^{3} - 4$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $- x^{3} - 4$ with respect to $x$ is $\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [- 4]$ .

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

Step 2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

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

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 2.4

Multiply $3$ by $- 1$ .

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

Step 2.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\ln (4 x^{2}) (- 3 x^{2} + 0) + (- x^{3} - 4) \frac{d}{d x} [\ln (4 x^{2})]$

Step 2.6

Add $- 3 x^{2}$ and $0$ .

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

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 $g (x) = 4 x^{2}$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u$ as $4 x^{2}$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $4 x^{2}$ .

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

Step 4

Differentiate.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Simplify terms.

Tap for more steps...

Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

Rewrite the expression.

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

Step 4.3

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

$\ln (4 x^{2}) (- 3 x^{2}) + (- x^{3} - 4) \frac{1}{x^{2}} (2 x)$

Step 4.4

Simplify terms.

Tap for more steps...

Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

Move $2$ to the left of $x$ .

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

Step 4.4.4

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

Tap for more steps...

Step 4.4.4.1

Factor $x$ out of $2 x$ .

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

Step 4.4.4.2

Cancel the common factors.

Tap for more steps...

Step 4.4.4.2.1

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

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

Step 4.4.4.2.2

Cancel the common factor.

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

Step 4.4.4.2.3

Rewrite the expression.

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

Combine terms.

Tap for more steps...

Step 5.2.1

Combine $\frac{2}{x}$ and $x^{3}$ .

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

Step 5.2.2

Cancel the common factor of $x^{3}$ and $x$ .

Tap for more steps...

Step 5.2.2.1

Factor $x$ out of $2 x^{3}$ .

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

Step 5.2.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.2.1

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

$\ln (4 x^{2}) (- 3 x^{2}) - \frac{x (2 x^{2})}{x^{1}} - 4 \frac{2}{x}$

Step 5.2.2.2.2

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

$\ln (4 x^{2}) (- 3 x^{2}) - \frac{x (2 x^{2})}{x \cdot 1} - 4 \frac{2}{x}$

Step 5.2.2.2.3

Cancel the common factor.

$\ln (4 x^{2}) (- 3 x^{2}) - \frac{x (2 x^{2})}{x \cdot 1} - 4 \frac{2}{x}$

Step 5.2.2.2.4

Rewrite the expression.

$\ln (4 x^{2}) (- 3 x^{2}) - \frac{2 x^{2}}{1} - 4 \frac{2}{x}$

Step 5.2.2.2.5

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

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

Step 5.2.3

Multiply $2$ by $- 1$ .

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

Step 5.2.4

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

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

Step 5.2.5

Multiply $- 4$ by $2$ .

$\ln (4 x^{2}) (- 3 x^{2}) - 2 x^{2} + \frac{- 8}{x}$

Step 5.2.6

Move the negative in front of the fraction.

$\ln (4 x^{2}) (- 3 x^{2}) - 2 x^{2} - \frac{8}{x}$

Step 5.3

Reorder terms.

$- 2 x^{2} - 3 x^{2} \ln (4 x^{2}) - \frac{8}{x}$