Calculus Examples

$y = x^{2} + 7.5 x + 4$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{2} + 7.5 x + 4)$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $x^{2} + 7.5 x + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [7.5 x] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [7.5 x] + \frac{d}{d x} [4]$

Step 3.1.2

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

$2 x + \frac{d}{d x} [7.5 x] + \frac{d}{d x} [4]$

Step 3.2

Evaluate $\frac{d}{d x} [7.5 x]$ .

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Step 3.2.1

Since $7.5$ is constant with respect to $x$ , the derivative of $7.5 x$ with respect to $x$ is $7.5 \frac{d}{d x} [x]$ .

$2 x + 7.5 \frac{d}{d x} [x] + \frac{d}{d x} [4]$

Step 3.2.2

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

$2 x + 7.5 \cdot 1 + \frac{d}{d x} [4]$

Step 3.2.3

Multiply $7.5$ by $1$ .

$2 x + 7.5 + \frac{d}{d x} [4]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$2 x + 7.5 + 0$

Step 3.3.2

Add $2 x + 7.5$ and $0$ .

$2 x + 7.5$

Step 4

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

$y' = 2 x + 7.5$

Step 5

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

$\frac{d y}{d x} = 2 x + 7.5$

Step 6

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

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Step 6.1

Subtract $7.5$ from both sides of the equation.

$2 x = - 7.5$

Step 6.2

Divide each term in $2 x = - 7.5$ by $2$ and simplify.

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Step 6.2.1

Divide each term in $2 x = - 7.5$ by $2$ .

$\frac{2 x}{2} = \frac{- 7.5}{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 7.5}{2}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7.5}{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $- 7.5$ by $2$ .

$x = - 3.75$

Step 7

Simplify ${(- 3.75)}^{2} + 7.5 (- 3.75) + 4$ .

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Step 7.1

Simplify each term.

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Step 7.1.1

Raise $- 3.75$ to the power of $2$ .

$y = 14.0625 + 7.5 (- 3.75) + 4$

Step 7.1.2

Multiply $7.5$ by $- 3.75$ .

$y = 14.0625 - 28.125 + 4$

Step 7.2

Simplify by adding and subtracting.

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Step 7.2.1

Subtract $28.125$ from $14.0625$ .

$y = - 14.0625 + 4$

Step 7.2.2

Add $- 14.0625$ and $4$ .

$y = - 10.0625$

Step 8

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

$(- 3.75, - 10.0625)$

Step 9