Calculus Examples

$y = 3 x^{2} + 4 x$

Step 1

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

$f (x) = 3 x^{2} + 4 x$

Step 2

Find the derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [4 x]$

Step 2.3

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

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Step 2.3.1

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

$6 x + 4 \frac{d}{d x} [x]$

Step 2.3.2

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

$6 x + 4 \cdot 1$

Step 2.3.3

Multiply $4$ by $1$ .

$6 x + 4$

Step 3

Set the derivative equal to $0$ then solve the equation $6 x + 4 = 0$ .

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Step 3.1

Subtract $4$ from both sides of the equation.

$6 x = - 4$

Step 3.2

Divide each term in $6 x = - 4$ by $6$ and simplify.

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Step 3.2.1

Divide each term in $6 x = - 4$ by $6$ .

$\frac{6 x}{6} = \frac{- 4}{6}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 4}{6}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{6}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $- 4$ and $6$ .

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Step 3.2.3.1.1

Factor $2$ out of $- 4$ .

$x = \frac{2 (- 2)}{6}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $2$ out of $6$ .

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

Step 3.2.3.1.2.2

Cancel the common factor.

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

Step 3.2.3.1.2.3

Rewrite the expression.

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

Step 3.2.3.2

Move the negative in front of the fraction.

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

Step 4

Solve the original function $f (x) = 3 x^{2} + 4 x$ at $x = - \frac{2}{3}$ .

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Step 4.1

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

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

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

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

Step 4.2.1.1.2

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

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

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

Step 4.2.1.5

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

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

Step 4.2.1.6

Cancel the common factor of $3$ .

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Step 4.2.1.6.1

Factor $3$ out of $9$ .

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

Step 4.2.1.6.2

Cancel the common factor.

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

Step 4.2.1.6.3

Rewrite the expression.

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

Step 4.2.1.7

Multiply $4 (- \frac{2}{3})$ .

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Step 4.2.1.7.1

Multiply $- 1$ by $4$ .

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

Step 4.2.1.7.2

Combine $- 4$ and $\frac{2}{3}$ .

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

Step 4.2.1.7.3

Multiply $- 4$ by $2$ .

$f (- \frac{2}{3}) = \frac{4}{3} + \frac{- 8}{3}$

Step 4.2.1.8

Move the negative in front of the fraction.

$f (- \frac{2}{3}) = \frac{4}{3} - \frac{8}{3}$

Step 4.2.2

Combine fractions.

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Step 4.2.2.1

Combine the numerators over the common denominator.

$f (- \frac{2}{3}) = \frac{4 - 8}{3}$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Subtract $8$ from $4$ .

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

Step 4.2.2.2.2

Move the negative in front of the fraction.

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

Step 4.2.3

The final answer is $- \frac{4}{3}$ .

$- \frac{4}{3}$

Step 5

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

$y = - \frac{4}{3}$

Step 6