Calculus Examples

$\frac{u}{u^{2} + 1}$

Step 1

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 $g (u) = u^{2} + 1$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Multiply $u^{2} + 1$ by $1$ .

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

Step 2.3

By the Sum Rule, the derivative of $u^{2} + 1$ with respect to $u$ is $\frac{d}{d u} [u^{2}] + \frac{d}{d u} [1]$ .

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

Step 2.4

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

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

Step 2.5

Since $1$ is constant with respect to $u$ , the derivative of $1$ with respect to $u$ is $0$ .

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $2 u$ and $0$ .

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

Step 2.6.2

Multiply $2$ by $- 1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

Subtract $2 u^{2}$ from $u^{2}$ .

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