Calculus Examples

$y = \frac{5}{8} x^{8} - 4 x^{6} + 5 x + 10$

Step 1

By the Sum Rule, the derivative of $\frac{5}{8} x^{8} - 4 x^{6} + 5 x + 10$ with respect to $x$ is $\frac{d}{d x} [\frac{5}{8} x^{8}] + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$ .

$\frac{d}{d x} [\frac{5}{8} x^{8}] + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 2

Evaluate $\frac{d}{d x} [\frac{5}{8} x^{8}]$ .

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Step 2.1

Since $\frac{5}{8}$ is constant with respect to $x$ , the derivative of $\frac{5}{8} x^{8}$ with respect to $x$ is $\frac{5}{8} \frac{d}{d x} [x^{8}]$ .

$\frac{5}{8} \frac{d}{d x} [x^{8}] + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$\frac{5}{8} (8 x^{7}) + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 2.3

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

$\frac{8 \cdot 5}{8} x^{7} + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 2.4

Multiply $8$ by $5$ .

$\frac{40}{8} x^{7} + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 2.5

Combine $\frac{40}{8}$ and $x^{7}$ .

$\frac{40 x^{7}}{8} + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 2.6

Cancel the common factor of $40$ and $8$ .

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Step 2.6.1

Factor $8$ out of $40 x^{7}$ .

$\frac{8 (5 x^{7})}{8} + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $8$ out of $8$ .

$\frac{8 (5 x^{7})}{8 (1)} + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 2.6.2.2

Cancel the common factor.

$\frac{8 (5 x^{7})}{8 \cdot 1} + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 2.6.2.3

Rewrite the expression.

$\frac{5 x^{7}}{1} + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 2.6.2.4

Divide $5 x^{7}$ by $1$ .

$5 x^{7} + \frac{d}{d x} [- 4 x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 3

Evaluate $\frac{d}{d x} [- 4 x^{6}]$ .

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Step 3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{6}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{6}]$ .

$5 x^{7} - 4 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$5 x^{7} - 4 (6 x^{5}) + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 3.3

Multiply $6$ by $- 4$ .

$5 x^{7} - 24 x^{5} + \frac{d}{d x} [5 x] + \frac{d}{d x} [10]$

Step 4

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 4.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 x^{7} - 24 x^{5} + 5 \frac{d}{d x} [x] + \frac{d}{d x} [10]$

Step 4.2

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

$5 x^{7} - 24 x^{5} + 5 \cdot 1 + \frac{d}{d x} [10]$

Step 4.3

Multiply $5$ by $1$ .

$5 x^{7} - 24 x^{5} + 5 + \frac{d}{d x} [10]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$5 x^{7} - 24 x^{5} + 5 + 0$

Step 5.2

Add $5 x^{7} - 24 x^{5} + 5$ and $0$ .

$5 x^{7} - 24 x^{5} + 5$