Calculus Examples

$f (x) = x^{3} - 3 x^{2} + 6 x + 1$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of $x^{3} - 3 x^{2} + 6 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [1]$

Step 1.1.1.2

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

$3 x^{2} + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [1]$

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

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

$3 x^{2} - 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [1]$

Step 1.1.2.2

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

$3 x^{2} - 3 (2 x) + \frac{d}{d x} [6 x] + \frac{d}{d x} [1]$

Step 1.1.2.3

Multiply $2$ by $- 3$ .

$3 x^{2} - 6 x + \frac{d}{d x} [6 x] + \frac{d}{d x} [1]$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

$3 x^{2} - 6 x + 6 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 1.1.3.2

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

$3 x^{2} - 6 x + 6 \cdot 1 + \frac{d}{d x} [1]$

Step 1.1.3.3

Multiply $6$ by $1$ .

$3 x^{2} - 6 x + 6 + \frac{d}{d x} [1]$

Step 1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.4.1

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

$3 x^{2} - 6 x + 6 + 0$

Step 1.1.4.2

Add $3 x^{2} - 6 x + 6$ and $0$ .

$f' (x) = 3 x^{2} - 6 x + 6$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 6 x + 6$ .

$3 x^{2} - 6 x + 6$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 6 x + 6 = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} - 6 x + 6 = 0$

Step 2.2

Factor $3$ out of $3 x^{2} - 6 x + 6$ .

Tap for more steps...

Step 2.2.1

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

$3 (x^{2}) - 6 x + 6 = 0$

Step 2.2.2

Factor $3$ out of $- 6 x$ .

$3 (x^{2}) + 3 (- 2 x) + 6 = 0$

Step 2.2.3

Factor $3$ out of $6$ .

$3 x^{2} + 3 (- 2 x) + 3 \cdot 2 = 0$

Step 2.2.4

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

$3 (x^{2} - 2 x) + 3 \cdot 2 = 0$

Step 2.2.5

Factor $3$ out of $3 (x^{2} - 2 x) + 3 \cdot 2$ .

$3 (x^{2} - 2 x + 2) = 0$

Step 2.3

Divide each term in $3 (x^{2} - 2 x + 2) = 0$ by $3$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $3 (x^{2} - 2 x + 2) = 0$ by $3$ .

$\frac{3 (x^{2} - 2 x + 2)}{3} = \frac{0}{3}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 (x^{2} - 2 x + 2)}{3} = \frac{0}{3}$

Step 2.3.2.1.2

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

$x^{2} - 2 x + 2 = \frac{0}{3}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Divide $0$ by $3$ .

$x^{2} - 2 x + 2 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5

Substitute the values $a = 1$ , $b = - 2$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 2.6

Simplify.

Tap for more steps...

Step 2.6.1

Simplify the numerator.

Tap for more steps...

Step 2.6.1.1

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 2.6.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 2.6.1.3

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 2.6.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 2.6.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 2.6.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 2.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i}{2}$

Step 2.6.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 2.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 2.7.1

Simplify the numerator.

Tap for more steps...

Step 2.7.1.1

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 2.7.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 2.7.1.3

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 2.7.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 2.7.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 2.7.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 2.7.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 2.7.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i}{2}$

Step 2.7.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 2.7.4

Change the $\pm$ to $+$ .

$x = 1 + i$

Step 2.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 2.8.1

Simplify the numerator.

Tap for more steps...

Step 2.8.1.1

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 2.8.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 2.8.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 2.8.1.3

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 2.8.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 2.8.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 2.8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 2.8.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 2.8.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 2.8.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 2.8.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i}{2}$

Step 2.8.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 2.8.4

Change the $\pm$ to $-$ .

$x = 1 - i$

Step 2.9

The final answer is the combination of both solutions.

$x = 1 + i, 1 - i$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found