Calculus Examples

$x + \frac{27}{2 x^{2}}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x + \frac{27}{2 x^{2}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{27}{2 x^{2}}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{27}{2 x^{2}}]$

Step 1.2

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

$1 + \frac{d}{d x} [\frac{27}{2 x^{2}}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{27}{2 x^{2}}]$ .

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Step 2.1

Since $\frac{27}{2}$ is constant with respect to $x$ , the derivative of $\frac{27}{2 x^{2}}$ with respect to $x$ is $\frac{27}{2} \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$1 + \frac{27}{2} \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2.2

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

$1 + \frac{27}{2} \frac{d}{d x} [{(x^{2})}^{- 1}]$

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 $x^{2}$ .

$1 + \frac{27}{2} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 2.3.2

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

$1 + \frac{27}{2} (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 2.3.3

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

$1 + \frac{27}{2} (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 2.4

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

$1 + \frac{27}{2} (- {(x^{2})}^{- 2} (2 x))$

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, ${(a^{m})}^{n} = a^{m n}$ .

$1 + \frac{27}{2} (- x^{2 \cdot - 2} (2 x))$

Step 2.5.2

Multiply $2$ by $- 2$ .

$1 + \frac{27}{2} (- x^{- 4} (2 x))$

Step 2.6

Multiply $2$ by $- 1$ .

$1 + \frac{27}{2} (- 2 x^{- 4} x)$

Step 2.7

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

$1 + \frac{27}{2} (- 2 (x^{1} x^{- 4}))$

Step 2.8

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

$1 + \frac{27}{2} (- 2 x^{1 - 4})$

Step 2.9

Subtract $4$ from $1$ .

$1 + \frac{27}{2} (- 2 x^{- 3})$

Step 2.10

Combine $- 2$ and $\frac{27}{2}$ .

$1 + \frac{- 2 \cdot 27}{2} x^{- 3}$

Step 2.11

Multiply $- 2$ by $27$ .

$1 + \frac{- 54}{2} x^{- 3}$

Step 2.12

Combine $\frac{- 54}{2}$ and $x^{- 3}$ .

$1 + \frac{- 54 x^{- 3}}{2}$

Step 2.13

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

$1 + \frac{- 54}{2 x^{3}}$

Step 2.14

Cancel the common factor of $- 54$ and $2$ .

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Step 2.14.1

Factor $2$ out of $- 54$ .

$1 + \frac{2 \cdot - 27}{2 x^{3}}$

Step 2.14.2

Cancel the common factors.

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Step 2.14.2.1

Factor $2$ out of $2 x^{3}$ .

$1 + \frac{2 \cdot - 27}{2 (x^{3})}$

Step 2.14.2.2

Cancel the common factor.

$1 + \frac{2 \cdot - 27}{2 x^{3}}$

Step 2.14.2.3

Rewrite the expression.

$1 + \frac{- 27}{x^{3}}$

Step 2.15

Move the negative in front of the fraction.

$1 - \frac{27}{x^{3}}$

Step 3

Reorder terms.

$- \frac{27}{x^{3}} + 1$