Calculus Examples

$f (x) = 2 x^{2} - x^{3}$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- x^{3}]$

Step 1.3

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

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Step 1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$4 x - \frac{d}{d x} [x^{3}]$

Step 1.3.2

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

$4 x - (3 x^{2})$

Step 1.3.3

Multiply $3$ by $- 1$ .

$4 x - 3 x^{2}$

Step 1.4

Reorder terms.

$f' (x) = - 3 x^{2} + 4 x$

Step 2

Find the second derivative.

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Step 2.1

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

$\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [4 x]$

Step 2.2

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

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Step 2.2.1

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

$- 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x]$

Step 2.2.2

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

$- 3 (2 x) + \frac{d}{d x} [4 x]$

Step 2.2.3

Multiply $2$ by $- 3$ .

$- 6 x + \frac{d}{d x} [4 x]$

Step 2.3

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

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Step 2.3.1

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

$- 6 x + 4 \frac{d}{d x} [x]$

Step 2.3.2

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

$- 6 x + 4 \cdot 1$

Step 2.3.3

Multiply $4$ by $1$ .

$f'' (x) = - 6 x + 4$

Step 3

Find the third derivative.

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Step 3.1

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

$\frac{d}{d x} [- 6 x] + \frac{d}{d x} [4]$

Step 3.2

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

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Step 3.2.1

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

$- 6 \frac{d}{d x} [x] + \frac{d}{d x} [4]$

Step 3.2.2

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

$- 6 \cdot 1 + \frac{d}{d x} [4]$

Step 3.2.3

Multiply $- 6$ by $1$ .

$- 6 + \frac{d}{d x} [4]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- 6 + 0$

Step 3.3.2

Add $- 6$ and $0$ .

$f''' (x) = - 6$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $- 6$ .

$- 6$