Calculus Examples

$y = 15 x^{- 2} - 3 x^{3} + 14 x$

Step 1

By the Sum Rule, the derivative of $15 x^{- 2} - 3 x^{3} + 14 x$ with respect to $x$ is $\frac{d}{d x} [15 x^{- 2}] + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [14 x]$ .

$\frac{d}{d x} [15 x^{- 2}] + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [14 x]$

Step 2

Evaluate $\frac{d}{d x} [15 x^{- 2}]$ .

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Step 2.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{- 2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{- 2}]$ .

$15 \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [14 x]$

Step 2.2

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

$15 (- 2 x^{- 3}) + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [14 x]$

Step 2.3

Multiply $- 2$ by $15$ .

$- 30 x^{- 3} + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [14 x]$

Step 3

Evaluate $\frac{d}{d x} [- 3 x^{3}]$ .

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{3}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{3}]$ .

$- 30 x^{- 3} - 3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [14 x]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$- 30 x^{- 3} - 3 (3 x^{2}) + \frac{d}{d x} [14 x]$

Step 3.3

Multiply $3$ by $- 3$ .

$- 30 x^{- 3} - 9 x^{2} + \frac{d}{d x} [14 x]$

Step 4

Evaluate $\frac{d}{d x} [14 x]$ .

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Step 4.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

$- 30 x^{- 3} - 9 x^{2} + 14 \frac{d}{d x} [x]$

Step 4.2

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

$- 30 x^{- 3} - 9 x^{2} + 14 \cdot 1$

Step 4.3

Multiply $14$ by $1$ .

$- 30 x^{- 3} - 9 x^{2} + 14$

Step 5

Simplify.

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Step 5.1

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

$- 30 \frac{1}{x^{3}} - 9 x^{2} + 14$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $- 30$ and $\frac{1}{x^{3}}$ .

$\frac{- 30}{x^{3}} - 9 x^{2} + 14$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{30}{x^{3}} - 9 x^{2} + 14$

Step 5.3

Reorder terms.

$- 9 x^{2} - \frac{30}{x^{3}} + 14$