Calculus Examples

$D (x) = 4.89 + 0.876 x - 0.1784 x^{2} + 0.0634 x^{3} - 0.00039 x^{4}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $4.89 + 0.876 x - 0.1784 x^{2} + 0.0634 x^{3} - 0.00039 x^{4}$ with respect to $x$ is $\frac{d}{d x} [4.89] + \frac{d}{d x} [0.876 x] + \frac{d}{d x} [- 0.1784 x^{2}] + \frac{d}{d x} [0.0634 x^{3}] + \frac{d}{d x} [- 0.00039 x^{4}]$ .

$\frac{d}{d x} [4.89] + \frac{d}{d x} [0.876 x] + \frac{d}{d x} [- 0.1784 x^{2}] + \frac{d}{d x} [0.0634 x^{3}] + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 1.2

Since $4.89$ is constant with respect to $x$ , the derivative of $4.89$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [0.876 x] + \frac{d}{d x} [- 0.1784 x^{2}] + \frac{d}{d x} [0.0634 x^{3}] + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 2

Evaluate $\frac{d}{d x} [0.876 x]$ .

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Step 2.1

Since $0.876$ is constant with respect to $x$ , the derivative of $0.876 x$ with respect to $x$ is $0.876 \frac{d}{d x} [x]$ .

$0 + 0.876 \frac{d}{d x} [x] + \frac{d}{d x} [- 0.1784 x^{2}] + \frac{d}{d x} [0.0634 x^{3}] + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 2.2

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

$0 + 0.876 \cdot 1 + \frac{d}{d x} [- 0.1784 x^{2}] + \frac{d}{d x} [0.0634 x^{3}] + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 2.3

Multiply $0.876$ by $1$ .

$0 + 0.876 + \frac{d}{d x} [- 0.1784 x^{2}] + \frac{d}{d x} [0.0634 x^{3}] + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 3

Evaluate $\frac{d}{d x} [- 0.1784 x^{2}]$ .

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Step 3.1

Since $- 0.1784$ is constant with respect to $x$ , the derivative of $- 0.1784 x^{2}$ with respect to $x$ is $- 0.1784 \frac{d}{d x} [x^{2}]$ .

$0 + 0.876 - 0.1784 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [0.0634 x^{3}] + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 3.2

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

$0 + 0.876 - 0.1784 (2 x) + \frac{d}{d x} [0.0634 x^{3}] + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 3.3

Multiply $2$ by $- 0.1784$ .

$0 + 0.876 - 0.3568 x + \frac{d}{d x} [0.0634 x^{3}] + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 4

Evaluate $\frac{d}{d x} [0.0634 x^{3}]$ .

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Step 4.1

Since $0.0634$ is constant with respect to $x$ , the derivative of $0.0634 x^{3}$ with respect to $x$ is $0.0634 \frac{d}{d x} [x^{3}]$ .

$0 + 0.876 - 0.3568 x + 0.0634 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 4.2

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

$0 + 0.876 - 0.3568 x + 0.0634 (3 x^{2}) + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 4.3

Multiply $3$ by $0.0634$ .

$0 + 0.876 - 0.3568 x + 0.1902 x^{2} + \frac{d}{d x} [- 0.00039 x^{4}]$

Step 5

Evaluate $\frac{d}{d x} [- 0.00039 x^{4}]$ .

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Step 5.1

Since $- 0.00039$ is constant with respect to $x$ , the derivative of $- 0.00039 x^{4}$ with respect to $x$ is $- 0.00039 \frac{d}{d x} [x^{4}]$ .

$0 + 0.876 - 0.3568 x + 0.1902 x^{2} - 0.00039 \frac{d}{d x} [x^{4}]$

Step 5.2

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

$0 + 0.876 - 0.3568 x + 0.1902 x^{2} - 0.00039 (4 x^{3})$

Step 5.3

Multiply $4$ by $- 0.00039$ .

$0 + 0.876 - 0.3568 x + 0.1902 x^{2} - 0.00156 x^{3}$

Step 6

Simplify.

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Step 6.1

Add $0$ and $0.876$ .

$0.876 - 0.3568 x + 0.1902 x^{2} - 0.00156 x^{3}$

Step 6.2

Reorder terms.

$- 0.00156 x^{3} + 0.1902 x^{2} - 0.3568 x + 0.876$