Calculus Examples

$- \frac{1}{3} (x^{7} + 2 x - 9)$

Step 1

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{1}{3} (x^{7} + 2 x - 9)$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [x^{7} + 2 x - 9]$ .

$- \frac{1}{3} \frac{d}{d x} [x^{7} + 2 x - 9]$

Step 2

By the Sum Rule, the derivative of $x^{7} + 2 x - 9$ with respect to $x$ is $\frac{d}{d x} [x^{7}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9]$ .

$- \frac{1}{3} (\frac{d}{d x} [x^{7}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9])$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$- \frac{1}{3} (7 x^{6} + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 9])$

Step 4

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

$- \frac{1}{3} (7 x^{6} + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 9])$

Step 5

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

$- \frac{1}{3} (7 x^{6} + 2 \cdot 1 + \frac{d}{d x} [- 9])$

Step 6

Multiply $2$ by $1$ .

$- \frac{1}{3} (7 x^{6} + 2 + \frac{d}{d x} [- 9])$

Step 7

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$- \frac{1}{3} (7 x^{6} + 2 + 0)$

Step 8

Add $7 x^{6} + 2$ and $0$ .

$- \frac{1}{3} (7 x^{6} + 2)$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$- \frac{1}{3} (7 x^{6}) - \frac{1}{3} \cdot 2$

Step 9.2

Combine terms.

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Step 9.2.1

Multiply $7$ by $- 1$ .

$- 7 (\frac{1}{3}) x^{6} - \frac{1}{3} \cdot 2$

Step 9.2.2

Combine $- 7$ and $\frac{1}{3}$ .

$\frac{- 7}{3} x^{6} - \frac{1}{3} \cdot 2$

Step 9.2.3

Combine $\frac{- 7}{3}$ and $x^{6}$ .

$\frac{- 7 x^{6}}{3} - \frac{1}{3} \cdot 2$

Step 9.2.4

Move the negative in front of the fraction.

$- \frac{7 x^{6}}{3} - \frac{1}{3} \cdot 2$

Step 9.2.5

Multiply $2$ by $- 1$ .

$- \frac{7 x^{6}}{3} - 2 (\frac{1}{3})$

Step 9.2.6

Combine $- 2$ and $\frac{1}{3}$ .

$- \frac{7 x^{6}}{3} + \frac{- 2}{3}$

Step 9.2.7

Move the negative in front of the fraction.

$- \frac{7 x^{6}}{3} - \frac{2}{3}$