Calculus Examples

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

Step 1

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

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

Step 2

Find the derivative.

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Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d x} [- 5 x^{2}]$ .

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Step 2.2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $- 5$ .

$- 10 x + \frac{d}{d x} [- 2 x]$

Step 2.3

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

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Step 2.3.1

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

$- 10 x - 2 \frac{d}{d x} [x]$

Step 2.3.2

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

$- 10 x - 2 \cdot 1$

Step 2.3.3

Multiply $- 2$ by $1$ .

$- 10 x - 2$

Step 3

Set the derivative equal to $0$ then solve the equation $- 10 x - 2 = 0$ .

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Step 3.1

Add $2$ to both sides of the equation.

$- 10 x = 2$

Step 3.2

Divide each term in $- 10 x = 2$ by $- 10$ and simplify.

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Step 3.2.1

Divide each term in $- 10 x = 2$ by $- 10$ .

$\frac{- 10 x}{- 10} = \frac{2}{- 10}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 10$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 10 x}{- 10} = \frac{2}{- 10}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{- 10}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $2$ and $- 10$ .

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Step 3.2.3.1.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{- 10}$

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 $- 10$ .

$x = \frac{2 \cdot 1}{2 \cdot - 5}$

Step 3.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 1}{2 \cdot - 5}$

Step 3.2.3.1.2.3

Rewrite the expression.

$x = \frac{1}{- 5}$

Step 3.2.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{5}$

Step 4

Solve the original function $f (x) = - 5 x^{2} - 2 x$ at $x = - \frac{1}{5}$ .

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Step 4.1

Replace the variable $x$ with $- \frac{1}{5}$ in the expression.

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

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 $- \frac{1}{5}$ .

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

Step 4.2.1.1.2

Apply the product rule to $\frac{1}{5}$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Multiply $\frac{1^{2}}{5^{2}}$ by $1$ .

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

Step 4.2.1.4

One to any power is one.

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

Step 4.2.1.5

Raise $5$ to the power of $2$ .

$f (- \frac{1}{5}) = - 5 (\frac{1}{25}) - 2 (- \frac{1}{5})$

Step 4.2.1.6

Cancel the common factor of $5$ .

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Step 4.2.1.6.1

Factor $5$ out of $- 5$ .

$f (- \frac{1}{5}) = 5 (- 1) (\frac{1}{25}) - 2 (- \frac{1}{5})$

Step 4.2.1.6.2

Factor $5$ out of $25$ .

$f (- \frac{1}{5}) = 5 \cdot (- 1 \frac{1}{5 \cdot 5}) - 2 (- \frac{1}{5})$

Step 4.2.1.6.3

Cancel the common factor.

$f (- \frac{1}{5}) = 5 \cdot (- 1 \frac{1}{5 \cdot 5}) - 2 (- \frac{1}{5})$

Step 4.2.1.6.4

Rewrite the expression.

$f (- \frac{1}{5}) = - 1 (\frac{1}{5}) - 2 (- \frac{1}{5})$

Step 4.2.1.7

Rewrite $- 1 (\frac{1}{5})$ as $- (\frac{1}{5})$ .

$f (- \frac{1}{5}) = - (\frac{1}{5}) - 2 (- \frac{1}{5})$

Step 4.2.1.8

Multiply $- 2 (- \frac{1}{5})$ .

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Step 4.2.1.8.1

Multiply $- 1$ by $- 2$ .

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

Step 4.2.1.8.2

Combine $2$ and $\frac{1}{5}$ .

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

Step 4.2.2

Combine fractions.

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Step 4.2.2.1

Combine the numerators over the common denominator.

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

Step 4.2.2.2

Add $- 1$ and $2$ .

$f (- \frac{1}{5}) = \frac{1}{5}$

Step 4.2.3

The final answer is $\frac{1}{5}$ .

$\frac{1}{5}$

Step 5

The horizontal tangent line on function $f (x) = - 5 x^{2} - 2 x$ is $y = \frac{1}{5}$ .

$y = \frac{1}{5}$

Step 6