Calculus Examples

$y = - \frac{1}{8} \cdot (x^{5} + 7 x - 4)$

Step 1

Since $- \frac{1}{8}$ is constant with respect to $x$ , the derivative of $- \frac{1}{8} (x^{5} + 7 x - 4)$ with respect to $x$ is $- \frac{1}{8} \frac{d}{d x} [x^{5} + 7 x - 4]$ .

$- \frac{1}{8} \frac{d}{d x} [x^{5} + 7 x - 4]$

Step 2

By the Sum Rule, the derivative of $x^{5} + 7 x - 4$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [7 x] + \frac{d}{d x} [- 4]$ .

$- \frac{1}{8} (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [7 x] + \frac{d}{d x} [- 4])$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$- \frac{1}{8} (5 x^{4} + \frac{d}{d x} [7 x] + \frac{d}{d x} [- 4])$

Step 4

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$- \frac{1}{8} (5 x^{4} + 7 \frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 5

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

$- \frac{1}{8} (5 x^{4} + 7 \cdot 1 + \frac{d}{d x} [- 4])$

Step 6

Multiply $7$ by $1$ .

$- \frac{1}{8} (5 x^{4} + 7 + \frac{d}{d x} [- 4])$

Step 7

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$- \frac{1}{8} (5 x^{4} + 7 + 0)$

Step 8

Add $5 x^{4} + 7$ and $0$ .

$- \frac{1}{8} (5 x^{4} + 7)$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$- \frac{1}{8} (5 x^{4}) - \frac{1}{8} \cdot 7$

Step 9.2

Combine terms.

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Step 9.2.1

Multiply $5$ by $- 1$ .

$- 5 (\frac{1}{8}) x^{4} - \frac{1}{8} \cdot 7$

Step 9.2.2

Combine $- 5$ and $\frac{1}{8}$ .

$\frac{- 5}{8} x^{4} - \frac{1}{8} \cdot 7$

Step 9.2.3

Combine $\frac{- 5}{8}$ and $x^{4}$ .

$\frac{- 5 x^{4}}{8} - \frac{1}{8} \cdot 7$

Step 9.2.4

Move the negative in front of the fraction.

$- \frac{5 x^{4}}{8} - \frac{1}{8} \cdot 7$

Step 9.2.5

Multiply $7$ by $- 1$ .

$- \frac{5 x^{4}}{8} - 7 (\frac{1}{8})$

Step 9.2.6

Combine $- 7$ and $\frac{1}{8}$ .

$- \frac{5 x^{4}}{8} + \frac{- 7}{8}$

Step 9.2.7

Move the negative in front of the fraction.

$- \frac{5 x^{4}}{8} - \frac{7}{8}$