Calculus Examples

$g (u) = \ln (\frac{u}{u + 3})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = \ln (u)$ and $g (u) = \frac{u}{u + 3}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{u}{u + 3}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d u} [\frac{u}{u + 3}]$

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d u} [\frac{u}{u + 3}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\frac{u}{u + 3}$ .

$\frac{1}{\frac{u}{u + 3}} \frac{d}{d u} [\frac{u}{u + 3}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{u}{u + 3}$ .

$1 \frac{u + 3}{u} \frac{d}{d u} [\frac{u}{u + 3}]$

Step 3

Multiply $\frac{u + 3}{u}$ by $1$ .

$\frac{u + 3}{u} \frac{d}{d u} [\frac{u}{u + 3}]$

Step 4

Differentiate using the Quotient Rule which states that $\frac{d}{d u} [\frac{f (u)}{g (u)}]$ is $\frac{g (u) \frac{d}{d u} [f (u)] - f (u) \frac{d}{d u} [g (u)]}{{g (u)}^{2}}$ where $f (u) = u$ and $g (u) = u + 3$ .

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

Multiply $u + 3$ by $1$ .

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Since $3$ is constant with respect to $u$ , the derivative of $3$ with respect to $u$ is $0$ .

$\frac{u + 3}{u} \cdot \frac{u + 3 - u (1 + 0)}{{(u + 3)}^{2}}$

Step 5.6

Simplify terms.

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Step 5.6.1

Add $1$ and $0$ .

$\frac{u + 3}{u} \cdot \frac{u + 3 - u \cdot 1}{{(u + 3)}^{2}}$

Step 5.6.2

Multiply $- 1$ by $1$ .

$\frac{u + 3}{u} \cdot \frac{u + 3 - u}{{(u + 3)}^{2}}$

Step 5.6.3

Subtract $u$ from $u$ .

$\frac{u + 3}{u} \cdot \frac{0 + 3}{{(u + 3)}^{2}}$

Step 5.6.4

Add $0$ and $3$ .

$\frac{u + 3}{u} \cdot \frac{3}{{(u + 3)}^{2}}$

Step 5.6.5

Multiply $\frac{u + 3}{u}$ by $\frac{3}{{(u + 3)}^{2}}$ .

$\frac{(u + 3) \cdot 3}{u {(u + 3)}^{2}}$

Step 5.6.6

Move $3$ to the left of $u + 3$ .

$\frac{3 \cdot (u + 3)}{u {(u + 3)}^{2}}$

Step 5.6.7

Cancel the common factor of $u + 3$ and ${(u + 3)}^{2}$ .

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Step 5.6.7.1

Factor $u + 3$ out of $3 (u + 3)$ .

$\frac{(u + 3) \cdot 3}{u {(u + 3)}^{2}}$

Step 5.6.7.2

Cancel the common factors.

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Step 5.6.7.2.1

Factor $u + 3$ out of $u {(u + 3)}^{2}$ .

$\frac{(u + 3) \cdot 3}{(u + 3) (u (u + 3))}$

Step 5.6.7.2.2

Cancel the common factor.

$\frac{(u + 3) \cdot 3}{(u + 3) (u (u + 3))}$

Step 5.6.7.2.3

Rewrite the expression.

$\frac{3}{u (u + 3)}$