Calculus Examples

$f (x) = 22 x^{5} - 17 x^{4} - 6 x^{3} + 10 x^{2} + 16 x - 2$

Step 1

By the Sum Rule, the derivative of $22 x^{5} - 17 x^{4} - 6 x^{3} + 10 x^{2} + 16 x - 2$ with respect to $x$ is $\frac{d}{d x} [22 x^{5}] + \frac{d}{d x} [- 17 x^{4}] + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [22 x^{5}] + \frac{d}{d x} [- 17 x^{4}] + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 2

Evaluate $\frac{d}{d x} [22 x^{5}]$ .

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Step 2.1

Since $22$ is constant with respect to $x$ , the derivative of $22 x^{5}$ with respect to $x$ is $22 \frac{d}{d x} [x^{5}]$ .

$22 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 17 x^{4}] + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 2.2

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

$22 (5 x^{4}) + \frac{d}{d x} [- 17 x^{4}] + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 2.3

Multiply $5$ by $22$ .

$110 x^{4} + \frac{d}{d x} [- 17 x^{4}] + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 3

Evaluate $\frac{d}{d x} [- 17 x^{4}]$ .

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Step 3.1

Since $- 17$ is constant with respect to $x$ , the derivative of $- 17 x^{4}$ with respect to $x$ is $- 17 \frac{d}{d x} [x^{4}]$ .

$110 x^{4} - 17 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$110 x^{4} - 17 (4 x^{3}) + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 3.3

Multiply $4$ by $- 17$ .

$110 x^{4} - 68 x^{3} + \frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 4

Evaluate $\frac{d}{d x} [- 6 x^{3}]$ .

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Step 4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{3}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{3}]$ .

$110 x^{4} - 68 x^{3} - 6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 4.2

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

$110 x^{4} - 68 x^{3} - 6 (3 x^{2}) + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 4.3

Multiply $3$ by $- 6$ .

$110 x^{4} - 68 x^{3} - 18 x^{2} + \frac{d}{d x} [10 x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 5

Evaluate $\frac{d}{d x} [10 x^{2}]$ .

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Step 5.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{2}$ with respect to $x$ is $10 \frac{d}{d x} [x^{2}]$ .

$110 x^{4} - 68 x^{3} - 18 x^{2} + 10 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 5.2

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

$110 x^{4} - 68 x^{3} - 18 x^{2} + 10 (2 x) + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 5.3

Multiply $2$ by $10$ .

$110 x^{4} - 68 x^{3} - 18 x^{2} + 20 x + \frac{d}{d x} [16 x] + \frac{d}{d x} [- 2]$

Step 6

Evaluate $\frac{d}{d x} [16 x]$ .

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Step 6.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x$ with respect to $x$ is $16 \frac{d}{d x} [x]$ .

$110 x^{4} - 68 x^{3} - 18 x^{2} + 20 x + 16 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 6.2

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

$110 x^{4} - 68 x^{3} - 18 x^{2} + 20 x + 16 \cdot 1 + \frac{d}{d x} [- 2]$

Step 6.3

Multiply $16$ by $1$ .

$110 x^{4} - 68 x^{3} - 18 x^{2} + 20 x + 16 + \frac{d}{d x} [- 2]$

Step 7

Differentiate using the Constant Rule.

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Step 7.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$110 x^{4} - 68 x^{3} - 18 x^{2} + 20 x + 16 + 0$

Step 7.2

Add $110 x^{4} - 68 x^{3} - 18 x^{2} + 20 x + 16$ and $0$ .

$110 x^{4} - 68 x^{3} - 18 x^{2} + 20 x + 16$