Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \ln (x + 4)$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\ln (x + 4)$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\ln (x + 4)$ .

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

Step 2

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 $g (x) = x + 4$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{2}$ as $x + 4$ .

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

Step 2.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

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

Step 2.3

Replace all occurrences of $u_{2}$ with $x + 4$ .

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

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

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

Step 3.2

Combine $\frac{2}{x + 4}$ and $\ln (x + 4)$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

$\frac{2 \ln (x + 4)}{x + 4} (1 + 0)$

Step 3.6

Simplify the expression.

Tap for more steps...

Step 3.6.1

Add $1$ and $0$ .

$\frac{2 \ln (x + 4)}{x + 4} \cdot 1$

Step 3.6.2

Multiply $\frac{2 \ln (x + 4)}{x + 4}$ by $1$ .

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

Step 4

Simplify $2 \ln (x + 4)$ by moving $2$ inside the logarithm.

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