Calculus Examples

$f (x) = 4 x^{2} + 2 x + 1$

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

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

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

Step 1.1.2

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

Tap for more steps...

Step 1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

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

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 $n = 2$ .

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

Step 1.1.2.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

$8 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

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 $n = 1$ .

$8 x + 2 \cdot 1 + \frac{d}{d x} [1]$

Step 1.1.3.3

Multiply $2$ by $1$ .

$8 x + 2 + \frac{d}{d x} [1]$

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 $0$ .

$8 x + 2 + 0$

Step 1.1.4.2

Add $8 x + 2$ and $0$ .

$f' (x) = 8 x + 2$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $8 x + 2$ .

$8 x + 2$

Step 2

Set the first derivative equal to $0$ then solve the equation $8 x + 2 = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$8 x + 2 = 0$

Step 2.2

Subtract $2$ from both sides of the equation.

$8 x = - 2$

Step 2.3

Divide each term in $8 x = - 2$ by $8$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $8 x = - 2$ by $8$ .

$\frac{8 x}{8} = \frac{- 2}{8}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 2}{8}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{8}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Cancel the common factor of $- 2$ and $8$ .

Tap for more steps...

Step 2.3.3.1.1

Factor $2$ out of $- 2$ .

$x = \frac{2 (- 1)}{8}$

Step 2.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.3.1.2.1

Factor $2$ out of $8$ .

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

Step 2.3.3.1.2.2

Cancel the common factor.

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

Step 2.3.3.1.2.3

Rewrite the expression.

$x = \frac{- 1}{4}$

Step 2.3.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

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

Evaluate $4 x^{2} + 2 x + 1$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = - \frac{1}{4}$ .

Tap for more steps...

Step 4.1.1

Substitute $- \frac{1}{4}$ for $x$ .

$4 {(- \frac{1}{4})}^{2} + 2 (- \frac{1}{4}) + 1$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Simplify each term.

Tap for more steps...

Step 4.1.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.1.2.1.1.1

Apply the product rule to $- \frac{1}{4}$ .

$4 ({(- 1)}^{2} {(\frac{1}{4})}^{2}) + 2 (- \frac{1}{4}) + 1$

Step 4.1.2.1.1.2

Apply the product rule to $\frac{1}{4}$ .

$4 ({(- 1)}^{2} \frac{1^{2}}{4^{2}}) + 2 (- \frac{1}{4}) + 1$

Step 4.1.2.1.2

Raise $- 1$ to the power of $2$ .

$4 (1 \frac{1^{2}}{4^{2}}) + 2 (- \frac{1}{4}) + 1$

Step 4.1.2.1.3

Multiply $\frac{1^{2}}{4^{2}}$ by $1$ .

$4 \frac{1^{2}}{4^{2}} + 2 (- \frac{1}{4}) + 1$

Step 4.1.2.1.4

One to any power is one.

$4 \frac{1}{4^{2}} + 2 (- \frac{1}{4}) + 1$

Step 4.1.2.1.5

Raise $4$ to the power of $2$ .

$4 (\frac{1}{16}) + 2 (- \frac{1}{4}) + 1$

Step 4.1.2.1.6

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.1.2.1.6.1

Factor $4$ out of $16$ .

$4 \frac{1}{4 (4)} + 2 (- \frac{1}{4}) + 1$

Step 4.1.2.1.6.2

Cancel the common factor.

$4 \frac{1}{4 \cdot 4} + 2 (- \frac{1}{4}) + 1$

Step 4.1.2.1.6.3

Rewrite the expression.

$\frac{1}{4} + 2 (- \frac{1}{4}) + 1$

Step 4.1.2.1.7

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.2.1.7.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$\frac{1}{4} + 2 (\frac{- 1}{4}) + 1$

Step 4.1.2.1.7.2

Factor $2$ out of $4$ .

$\frac{1}{4} + 2 \frac{- 1}{2 (2)} + 1$

Step 4.1.2.1.7.3

Cancel the common factor.

$\frac{1}{4} + 2 \frac{- 1}{2 \cdot 2} + 1$

Step 4.1.2.1.7.4

Rewrite the expression.

$\frac{1}{4} + \frac{- 1}{2} + 1$

Step 4.1.2.1.8

Move the negative in front of the fraction.

$\frac{1}{4} - \frac{1}{2} + 1$

Step 4.1.2.2

Find the common denominator.

Tap for more steps...

Step 4.1.2.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\frac{1}{4} - (\frac{1}{2} \cdot \frac{2}{2}) + 1$

Step 4.1.2.2.2

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\frac{1}{4} - \frac{2}{2 \cdot 2} + 1$

Step 4.1.2.2.3

Write $1$ as a fraction with denominator $1$ .

$\frac{1}{4} - \frac{2}{2 \cdot 2} + \frac{1}{1}$

Step 4.1.2.2.4

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

$\frac{1}{4} - \frac{2}{2 \cdot 2} + \frac{1}{1} \cdot \frac{4}{4}$

Step 4.1.2.2.5

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

$\frac{1}{4} - \frac{2}{2 \cdot 2} + \frac{4}{4}$

Step 4.1.2.2.6

Multiply $2$ by $2$ .

$\frac{1}{4} - \frac{2}{4} + \frac{4}{4}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$\frac{1 - 2 + 4}{4}$

Step 4.1.2.4

Simplify by adding and subtracting.

Tap for more steps...

Step 4.1.2.4.1

Subtract $2$ from $1$ .

$\frac{- 1 + 4}{4}$

Step 4.1.2.4.2

Add $- 1$ and $4$ .

$\frac{3}{4}$

Step 4.2

List all of the points.

$(- \frac{1}{4}, \frac{3}{4})$

Step 5