Calculus Examples

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

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{2} + 5 x - 7)$

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Differentiate the right side of the equation.

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

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

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

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

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

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

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

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

$2 x + 5 \cdot 1 + \frac{d}{d x} [- 7]$

Multiply $5$ by $1$ .

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

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$2 x + 5 + 0$

Add $2 x + 5$ and $0$ .

$2 x + 5$

Reform the equation by setting the left side equal to the right side.

$y' = 2 x + 5$

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 2 x + 5$

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Rewrite the equation as $2 x + 5 = 0$ .

$2 x + 5 = 0$

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = - 5$ by $2$ .

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Solve for $y$ .

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Simplify each term.

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Apply the product rule to $- \frac{5}{2}$ .

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

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

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

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

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

Apply the product rule to $\frac{5}{2}$ .

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

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

$y = \frac{25}{2^{2}} + 5 (- \frac{5}{2}) - 7$

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

$y = \frac{25}{4} + 5 (- \frac{5}{2}) - 7$

Simplify $5 (- \frac{5}{2})$ .

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Multiply $- 1$ by $5$ .

$y = \frac{25}{4} - 5 (\frac{5}{2}) - 7$

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

$y = \frac{25}{4} + \frac{- 5}{1} \cdot \frac{5}{2} - 7$

Multiply $\frac{- 5}{1}$ and $\frac{5}{2}$ .

$y = \frac{25}{4} + \frac{- 5 \cdot 5}{2} - 7$

Multiply $- 5$ by $5$ .

$y = \frac{25}{4} + \frac{- 25}{2} - 7$

Move the negative in front of the fraction.

$y = \frac{25}{4} - \frac{25}{2} - 7$

To write $\frac{25}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{25}{4} - \frac{25}{2} \cdot \frac{2}{2} - 7$

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

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Combine.

$y = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} - 7$

Multiply $2$ by $2$ .

$y = \frac{25}{4} - \frac{25 \cdot 2}{4} - 7$

Combine the numerators over the common denominator.

$y = \frac{25 - (25 \cdot 2)}{4} - 7$

Simplify the numerator.

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Multiply $25$ by $2$ .

$y = \frac{25 - 1 \cdot 50}{4} - 7$

Multiply $- 1$ by $50$ .

$y = \frac{25 - 50}{4} - 7$

Subtract $50$ from $25$ .

$y = \frac{- 25}{4} - 7$

Move the negative in front of the fraction.

$y = - \frac{25}{4} - 7$

To write $- \frac{- 7}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - \frac{25}{4} + \frac{- 7}{1} \cdot \frac{4}{4}$

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

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Combine.

$y = - \frac{25}{4} + \frac{- 7 \cdot 4}{1 \cdot 4}$

Multiply $4$ by $1$ .

$y = - \frac{25}{4} + \frac{- 7 \cdot 4}{4}$

Combine the numerators over the common denominator.

$y = \frac{- 1 \cdot 25 - 7 \cdot 4}{4}$

Simplify the numerator.

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Multiply $- 1$ by $25$ .

$y = \frac{- 25 - 7 \cdot 4}{4}$

Multiply $- 7$ by $4$ .

$y = \frac{- 25 - 28}{4}$

Subtract $28$ from $- 25$ .

$y = \frac{- 53}{4}$

Move the negative in front of the fraction.

$y = - \frac{53}{4}$

Find the points where $\frac{d y}{d x} = 0$ .

$(- \frac{5}{2}, - \frac{53}{4})$

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