Calculus Examples

$y = x^{2} + 3 x$

Step 1

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

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

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

$2 x + 3 \cdot 1$

Step 2.2.3

Multiply $3$ by $1$ .

$2 x + 3$

Step 3

Set the derivative equal to $0$ then solve the equation $2 x + 3 = 0$ .

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Step 3.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 3.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 4

Solve the original function $f (x) = x^{2} + 3 x$ at $x = - \frac{3}{2}$ .

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Step 4.1

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

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

Step 4.2.1.1.2

Apply the product rule to $\frac{3}{2}$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

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

Step 4.2.1.4

Raise $3$ to the power of $2$ .

$f (- \frac{3}{2}) = \frac{9}{2^{2}} + 3 (- \frac{3}{2})$

Step 4.2.1.5

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

$f (- \frac{3}{2}) = \frac{9}{4} + 3 (- \frac{3}{2})$

Step 4.2.1.6

Multiply $3 (- \frac{3}{2})$ .

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Step 4.2.1.6.1

Multiply $- 1$ by $3$ .

$f (- \frac{3}{2}) = \frac{9}{4} - 3 (\frac{3}{2})$

Step 4.2.1.6.2

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

$f (- \frac{3}{2}) = \frac{9}{4} + \frac{- 3 \cdot 3}{2}$

Step 4.2.1.6.3

Multiply $- 3$ by $3$ .

$f (- \frac{3}{2}) = \frac{9}{4} + \frac{- 9}{2}$

Step 4.2.1.7

Move the negative in front of the fraction.

$f (- \frac{3}{2}) = \frac{9}{4} - \frac{9}{2}$

Step 4.2.2

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{3}{2}) = \frac{9}{4} - \frac{9}{2} \cdot \frac{2}{2}$

Step 4.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$f (- \frac{3}{2}) = \frac{9}{4} - \frac{9 \cdot 2}{2 \cdot 2}$

Step 4.2.3.2

Multiply $2$ by $2$ .

$f (- \frac{3}{2}) = \frac{9}{4} - \frac{9 \cdot 2}{4}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (- \frac{3}{2}) = \frac{9 - 9 \cdot 2}{4}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 9$ by $2$ .

$f (- \frac{3}{2}) = \frac{9 - 18}{4}$

Step 4.2.5.2

Subtract $18$ from $9$ .

$f (- \frac{3}{2}) = \frac{- 9}{4}$

Step 4.2.6

Move the negative in front of the fraction.

$f (- \frac{3}{2}) = - \frac{9}{4}$

Step 4.2.7

The final answer is $- \frac{9}{4}$ .

$- \frac{9}{4}$

Step 5

The horizontal tangent line on function $f (x) = x^{2} + 3 x$ is $y = - \frac{9}{4}$ .

$y = - \frac{9}{4}$

Step 6