Calculus Examples

$y = 2 x^{2} + 7 x - 5$

Step 1

Set $y$ as a function of $x$ .

$f (x) = 2 x^{2} + 7 x - 5$

Step 2

Find the derivative.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $2 x^{2} + 7 x - 5$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [7 x] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [7 x] + \frac{d}{d x} [- 5]$

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [7 x] + \frac{d}{d x} [- 5]$

Step 2.2.2

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

$2 (2 x) + \frac{d}{d x} [7 x] + \frac{d}{d x} [- 5]$

Step 2.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [7 x] + \frac{d}{d x} [- 5]$

Step 2.3

Evaluate $\frac{d}{d x} [7 x]$ .

Tap for more steps...

Step 2.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$4 x + 7 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]$

Step 2.3.2

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

$4 x + 7 \cdot 1 + \frac{d}{d x} [- 5]$

Step 2.3.3

Multiply $7$ by $1$ .

$4 x + 7 + \frac{d}{d x} [- 5]$

Step 2.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$4 x + 7 + 0$

Step 2.4.2

Add $4 x + 7$ and $0$ .

$4 x + 7$

Step 3

Set the derivative equal to $0$ then solve the equation $4 x + 7 = 0$ .

Tap for more steps...

Step 3.1

Subtract $7$ from both sides of the equation.

$4 x = - 7$

Step 3.2

Divide each term in $4 x = - 7$ by $4$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $4 x = - 7$ by $4$ .

$\frac{4 x}{4} = \frac{- 7}{4}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 7}{4}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{4}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Move the negative in front of the fraction.

$x = - \frac{7}{4}$

Step 4

Solve the original function $f (x) = 2 x^{2} + 7 x - 5$ at $x = - \frac{7}{4}$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $- \frac{7}{4}$ in the expression.

$f (- \frac{7}{4}) = 2 {(- \frac{7}{4})}^{2} + 7 (- \frac{7}{4}) - 5$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

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

Tap for more steps...

Step 4.2.1.1.1

Apply the product rule to $- \frac{7}{4}$ .

$f (- \frac{7}{4}) = 2 ({(- 1)}^{2} {(\frac{7}{4})}^{2}) + 7 (- \frac{7}{4}) - 5$

Step 4.2.1.1.2

Apply the product rule to $\frac{7}{4}$ .

$f (- \frac{7}{4}) = 2 ({(- 1)}^{2} (\frac{7^{2}}{4^{2}})) + 7 (- \frac{7}{4}) - 5$

Step 4.2.1.2

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

$f (- \frac{7}{4}) = 2 (1 (\frac{7^{2}}{4^{2}})) + 7 (- \frac{7}{4}) - 5$

Step 4.2.1.3

Multiply $\frac{7^{2}}{4^{2}}$ by $1$ .

$f (- \frac{7}{4}) = 2 (\frac{7^{2}}{4^{2}}) + 7 (- \frac{7}{4}) - 5$

Step 4.2.1.4

Raise $7$ to the power of $2$ .

$f (- \frac{7}{4}) = 2 (\frac{49}{4^{2}}) + 7 (- \frac{7}{4}) - 5$

Step 4.2.1.5

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

$f (- \frac{7}{4}) = 2 (\frac{49}{16}) + 7 (- \frac{7}{4}) - 5$

Step 4.2.1.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.1.6.1

Factor $2$ out of $16$ .

$f (- \frac{7}{4}) = 2 (\frac{49}{2 (8)}) + 7 (- \frac{7}{4}) - 5$

Step 4.2.1.6.2

Cancel the common factor.

$f (- \frac{7}{4}) = 2 (\frac{49}{2 \cdot 8}) + 7 (- \frac{7}{4}) - 5$

Step 4.2.1.6.3

Rewrite the expression.

$f (- \frac{7}{4}) = \frac{49}{8} + 7 (- \frac{7}{4}) - 5$

Step 4.2.1.7

Multiply $7 (- \frac{7}{4})$ .

Tap for more steps...

Step 4.2.1.7.1

Multiply $- 1$ by $7$ .

$f (- \frac{7}{4}) = \frac{49}{8} - 7 (\frac{7}{4}) - 5$

Step 4.2.1.7.2

Combine $- 7$ and $\frac{7}{4}$ .

$f (- \frac{7}{4}) = \frac{49}{8} + \frac{- 7 \cdot 7}{4} - 5$

Step 4.2.1.7.3

Multiply $- 7$ by $7$ .

$f (- \frac{7}{4}) = \frac{49}{8} + \frac{- 49}{4} - 5$

Step 4.2.1.8

Move the negative in front of the fraction.

$f (- \frac{7}{4}) = \frac{49}{8} - \frac{49}{4} - 5$

Step 4.2.2

Find the common denominator.

Tap for more steps...

Step 4.2.2.1

Multiply $\frac{49}{4}$ by $\frac{2}{2}$ .

$f (- \frac{7}{4}) = \frac{49}{8} - (\frac{49}{4} \cdot \frac{2}{2}) - 5$

Step 4.2.2.2

Multiply $\frac{49}{4}$ by $\frac{2}{2}$ .

$f (- \frac{7}{4}) = \frac{49}{8} - \frac{49 \cdot 2}{4 \cdot 2} - 5$

Step 4.2.2.3

Write $- 5$ as a fraction with denominator $1$ .

$f (- \frac{7}{4}) = \frac{49}{8} - \frac{49 \cdot 2}{4 \cdot 2} + \frac{- 5}{1}$

Step 4.2.2.4

Multiply $\frac{- 5}{1}$ by $\frac{8}{8}$ .

$f (- \frac{7}{4}) = \frac{49}{8} - \frac{49 \cdot 2}{4 \cdot 2} + \frac{- 5}{1} \cdot \frac{8}{8}$

Step 4.2.2.5

Multiply $\frac{- 5}{1}$ by $\frac{8}{8}$ .

$f (- \frac{7}{4}) = \frac{49}{8} - \frac{49 \cdot 2}{4 \cdot 2} + \frac{- 5 \cdot 8}{8}$

Step 4.2.2.6

Reorder the factors of $4 \cdot 2$ .

$f (- \frac{7}{4}) = \frac{49}{8} - \frac{49 \cdot 2}{2 \cdot 4} + \frac{- 5 \cdot 8}{8}$

Step 4.2.2.7

Multiply $2$ by $4$ .

$f (- \frac{7}{4}) = \frac{49}{8} - \frac{49 \cdot 2}{8} + \frac{- 5 \cdot 8}{8}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (- \frac{7}{4}) = \frac{49 - 49 \cdot 2 - 5 \cdot 8}{8}$

Step 4.2.4

Simplify each term.

Tap for more steps...

Step 4.2.4.1

Multiply $- 49$ by $2$ .

$f (- \frac{7}{4}) = \frac{49 - 98 - 5 \cdot 8}{8}$

Step 4.2.4.2

Multiply $- 5$ by $8$ .

$f (- \frac{7}{4}) = \frac{49 - 98 - 40}{8}$

Step 4.2.5

Simplify the expression.

Tap for more steps...

Step 4.2.5.1

Subtract $98$ from $49$ .

$f (- \frac{7}{4}) = \frac{- 49 - 40}{8}$

Step 4.2.5.2

Subtract $40$ from $- 49$ .

$f (- \frac{7}{4}) = \frac{- 89}{8}$

Step 4.2.5.3

Move the negative in front of the fraction.

$f (- \frac{7}{4}) = - \frac{89}{8}$

Step 4.2.6

The final answer is $- \frac{89}{8}$ .

$- \frac{89}{8}$

Step 5

The horizontal tangent line on function $f (x) = 2 x^{2} + 7 x - 5$ is $y = - \frac{89}{8}$ .

$y = - \frac{89}{8}$

Step 6