Calculus Examples

$f (x) = \frac{x^{4}}{4} - \frac{x^{3}}{3} + \frac{x^{2}}{2} + x - 1$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $\frac{x^{4}}{4} - \frac{x^{3}}{3} + \frac{x^{2}}{2} + x - 1$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{4}}{4}] + \frac{d}{d x} [- \frac{x^{3}}{3}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [\frac{x^{4}}{4}] + \frac{d}{d x} [- \frac{x^{3}}{3}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x^{4}}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{x^{3}}{3}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.2.2

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

$\frac{1}{4} (4 x^{3}) + \frac{d}{d x} [- \frac{x^{3}}{3}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.2.3

Combine $4$ and $\frac{1}{4}$ .

$\frac{4}{4} x^{3} + \frac{d}{d x} [- \frac{x^{3}}{3}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.2.4

Combine $\frac{4}{4}$ and $x^{3}$ .

$\frac{4 x^{3}}{4} + \frac{d}{d x} [- \frac{x^{3}}{3}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.2.5

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.5.1

Cancel the common factor.

$\frac{4 x^{3}}{4} + \frac{d}{d x} [- \frac{x^{3}}{3}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.2.5.2

Divide $x^{3}$ by $1$ .

$x^{3} + \frac{d}{d x} [- \frac{x^{3}}{3}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$x^{3} - \frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.2

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

$x^{3} - \frac{1}{3} (3 x^{2}) + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.3

Multiply $3$ by $- 1$ .

$x^{3} - 3 (\frac{1}{3}) x^{2} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.4

Combine $- 3$ and $\frac{1}{3}$ .

$x^{3} + \frac{- 3}{3} x^{2} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.5

Combine $\frac{- 3}{3}$ and $x^{2}$ .

$x^{3} + \frac{- 3 x^{2}}{3} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.6

Cancel the common factor of $- 3$ and $3$ .

Tap for more steps...

Step 1.3.6.1

Factor $3$ out of $- 3 x^{2}$ .

$x^{3} + \frac{3 (- x^{2})}{3} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.6.2

Cancel the common factors.

Tap for more steps...

Step 1.3.6.2.1

Factor $3$ out of $3$ .

$x^{3} + \frac{3 (- x^{2})}{3 (1)} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.6.2.2

Cancel the common factor.

$x^{3} + \frac{3 (- x^{2})}{3 \cdot 1} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.6.2.3

Rewrite the expression.

$x^{3} + \frac{- x^{2}}{1} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.6.2.4

Divide $- x^{2}$ by $1$ .

$x^{3} - x^{2} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.4

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

Tap for more steps...

Step 1.4.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{2}]$ .

$x^{3} - x^{2} + \frac{1}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.4.2

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

$x^{3} - x^{2} + \frac{1}{2} (2 x) + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.4.3

Combine $2$ and $\frac{1}{2}$ .

$x^{3} - x^{2} + \frac{2}{2} x + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.4.4

Combine $\frac{2}{2}$ and $x$ .

$x^{3} - x^{2} + \frac{2 x}{2} + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.4.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.5.1

Cancel the common factor.

$x^{3} - x^{2} + \frac{2 x}{2} + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.4.5.2

Divide $x$ by $1$ .

$x^{3} - x^{2} + x + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.5

Differentiate.

Tap for more steps...

Step 1.5.1

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

$x^{3} - x^{2} + x + 1 + \frac{d}{d x} [- 1]$

Step 1.5.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$x^{3} - x^{2} + x + 1 + 0$

Step 1.5.3

Add $x^{3} - x^{2} + x + 1$ and $0$ .

$f' (x) = x^{3} - x^{2} + x + 1$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Differentiate.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $x^{3} - x^{2} + x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.1.2

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

$3 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.2.2

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

$3 x^{2} - (2 x) + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.2.3

Multiply $2$ by $- 1$ .

$3 x^{2} - 2 x + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.3

Differentiate.

Tap for more steps...

Step 2.3.1

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

$3 x^{2} - 2 x + 1 + \frac{d}{d x} [1]$

Step 2.3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 x^{2} - 2 x + 1 + 0$

Step 2.3.3

Add $3 x^{2} - 2 x + 1$ and $0$ .

$f'' (x) = 3 x^{2} - 2 x + 1$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 2 x + 1$ .

$3 x^{2} - 2 x + 1$