Calculus Examples

$y = \frac{1}{5 x^{2}} + \frac{1}{5 x}$

Step 1

By the Sum Rule, the derivative of $\frac{1}{5 x^{2}} + \frac{1}{5 x}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{5 x^{2}}] + \frac{d}{d x} [\frac{1}{5 x}]$ .

$\frac{d}{d x} [\frac{1}{5 x^{2}}] + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{5 x^{2}}]$ .

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Step 2.1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{1}{5 x^{2}}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$\frac{1}{5} \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$\frac{1}{5} \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 2.3.1

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

$\frac{1}{5} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.3.2

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

$\frac{1}{5} (- u^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.3.3

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

$\frac{1}{5} (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.4

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

$\frac{1}{5} (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.5

Multiply the exponents in ${(x^{2})}^{- 2}$ .

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Step 2.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{5} (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.5.2

Multiply $2$ by $- 2$ .

$\frac{1}{5} (- x^{- 4} (2 x)) + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.6

Multiply $2$ by $- 1$ .

$\frac{1}{5} (- 2 x^{- 4} x) + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.7

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

$\frac{1}{5} (- 2 (x^{1} x^{- 4})) + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.8

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

$\frac{1}{5} (- 2 x^{1 - 4}) + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.9

Subtract $4$ from $1$ .

$\frac{1}{5} (- 2 x^{- 3}) + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.10

Combine $- 2$ and $\frac{1}{5}$ .

$\frac{- 2}{5} x^{- 3} + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.11

Combine $\frac{- 2}{5}$ and $x^{- 3}$ .

$\frac{- 2 x^{- 3}}{5} + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.12

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 2}{5 x^{3}} + \frac{d}{d x} [\frac{1}{5 x}]$

Step 2.13

Move the negative in front of the fraction.

$- \frac{2}{5 x^{3}} + \frac{d}{d x} [\frac{1}{5 x}]$

Step 3

Evaluate $\frac{d}{d x} [\frac{1}{5 x}]$ .

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Step 3.1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{1}{5 x}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [\frac{1}{x}]$ .

$- \frac{2}{5 x^{3}} + \frac{1}{5} \frac{d}{d x} [\frac{1}{x}]$

Step 3.2

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

$- \frac{2}{5 x^{3}} + \frac{1}{5} \frac{d}{d x} [x^{- 1}]$

Step 3.3

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

$- \frac{2}{5 x^{3}} + \frac{1}{5} (- x^{- 2})$

Step 3.4

Combine $\frac{1}{5}$ and $x^{- 2}$ .

$- \frac{2}{5 x^{3}} - \frac{x^{- 2}}{5}$

Step 3.5

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{2}{5 x^{3}} - \frac{1}{5 x^{2}}$

Step 4

Reorder terms.

$- \frac{1}{5 x^{2}} - \frac{2}{5 x^{3}}$