Calculus Examples

$12 x^{3} - 117 x^{2} \cdot (432 x) + 6$

Step 1

By the Sum Rule, the derivative of $12 x^{3} - 117 x^{2} \cdot (432 x) + 6$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [- 117 x^{2} \cdot (432 x)] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [- 117 x^{2} \cdot (432 x)] + \frac{d}{d x} [6]$

Step 2

Evaluate $\frac{d}{d x} [12 x^{3}]$ .

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Step 2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{3}$ with respect to $x$ is $12 \frac{d}{d x} [x^{3}]$ .

$12 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 117 x^{2} \cdot (432 x)] + \frac{d}{d x} [6]$

Step 2.2

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

$12 (3 x^{2}) + \frac{d}{d x} [- 117 x^{2} \cdot (432 x)] + \frac{d}{d x} [6]$

Step 2.3

Multiply $3$ by $12$ .

$36 x^{2} + \frac{d}{d x} [- 117 x^{2} \cdot (432 x)] + \frac{d}{d x} [6]$

Step 3

Evaluate $\frac{d}{d x} [- 117 x^{2} \cdot (432 x)]$ .

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Step 3.1

Multiply $432$ by $- 117$ .

$36 x^{2} + \frac{d}{d x} [- 50544 x^{2} \cdot x] + \frac{d}{d x} [6]$

Step 3.2

Raise $x$ to the power of $1$ .

$36 x^{2} + \frac{d}{d x} [- 50544 (x^{1} x^{2})] + \frac{d}{d x} [6]$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$36 x^{2} + \frac{d}{d x} [- 50544 x^{1 + 2}] + \frac{d}{d x} [6]$

Step 3.4

Add $1$ and $2$ .

$36 x^{2} + \frac{d}{d x} [- 50544 x^{3}] + \frac{d}{d x} [6]$

Step 3.5

Since $- 50544$ is constant with respect to $x$ , the derivative of $- 50544 x^{3}$ with respect to $x$ is $- 50544 \frac{d}{d x} [x^{3}]$ .

$36 x^{2} - 50544 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [6]$

Step 3.6

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

$36 x^{2} - 50544 (3 x^{2}) + \frac{d}{d x} [6]$

Step 3.7

Multiply $3$ by $- 50544$ .

$36 x^{2} - 151632 x^{2} + \frac{d}{d x} [6]$

Step 4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$36 x^{2} - 151632 x^{2} + 0$

Step 5

Combine terms.

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Step 5.1

Subtract $151632 x^{2}$ from $36 x^{2}$ .

$- 151596 x^{2} + 0$

Step 5.2

Add $- 151596 x^{2}$ and $0$ .

$- 151596 x^{2}$