Calculus Examples

$2 x - 27 x^{- 2}$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

$2 - 27 (- 2 x^{- 3})$

Step 3.3

Multiply $- 2$ by $- 27$ .

$2 + 54 x^{- 3}$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$54 x^{- 3} + 2$

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

Combine $54$ and $\frac{1}{x^{3}}$ .

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