Calculus Examples

$y = 2 x^{3} - 3 x^{2} - 12 x + 20$

Step 1

By the Sum Rule, the derivative of $2 x^{3} - 3 x^{2} - 12 x + 20$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [20]$ .

$\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [20]$

Step 2

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [20]$

Step 2.2

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

$2 (3 x^{2}) + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [20]$

Step 2.3

Multiply $3$ by $2$ .

$6 x^{2} + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [20]$

Step 3

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$6 x^{2} - 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [20]$

Step 3.2

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

$6 x^{2} - 3 (2 x) + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [20]$

Step 3.3

Multiply $2$ by $- 3$ .

$6 x^{2} - 6 x + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [20]$

Step 4

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

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Step 4.1

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

$6 x^{2} - 6 x - 12 \frac{d}{d x} [x] + \frac{d}{d x} [20]$

Step 4.2

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

$6 x^{2} - 6 x - 12 \cdot 1 + \frac{d}{d x} [20]$

Step 4.3

Multiply $- 12$ by $1$ .

$6 x^{2} - 6 x - 12 + \frac{d}{d x} [20]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$6 x^{2} - 6 x - 12 + 0$

Step 5.2

Add $6 x^{2} - 6 x - 12$ and $0$ .

$6 x^{2} - 6 x - 12$