Calculus Examples

$y = u + \frac{1}{u}$ ; $u = 5$

Differentiate.

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

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

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

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

Evaluate $\frac{d}{d u} [\frac{1}{u}]$ .

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Rewrite $\frac{1}{u}$ as $u^{- 1}$ .

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

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

$1 - u^{- 2}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Reorder terms.

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

Evaluate the derivative at $u = 5$ .

$- \frac{1}{{(5)}^{2}} + 1$

Simplify.

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Raise $5$ to the power of $2$ .

$- \frac{1}{25} + 1$

Simplify the expression.

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Write $1$ as a fraction with a common denominator.

$- \frac{1}{25} + \frac{25}{25}$

Combine the numerators over the common denominator.

$\frac{- 1 + 25}{25}$

Add $- 1$ and $25$ .

$\frac{24}{25}$