Calculus Examples

$s = 180 A - 0.3 A^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d A} (s) = \frac{d}{d A} (180 A - 0.3 A^{3})$

Step 2

Since $s$ is constant with respect to $A$ , the derivative of $s$ with respect to $A$ is $0$ .

$0$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $180 A - 0.3 A^{3}$ with respect to $A$ is $\frac{d}{d A} [180 A] + \frac{d}{d A} [- 0.3 A^{3}]$ .

$\frac{d}{d A} [180 A] + \frac{d}{d A} [- 0.3 A^{3}]$

Step 3.2

Evaluate $\frac{d}{d A} [180 A]$ .

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Step 3.2.1

Since $180$ is constant with respect to $A$ , the derivative of $180 A$ with respect to $A$ is $180 \frac{d}{d A} [A]$ .

$180 \frac{d}{d A} [A] + \frac{d}{d A} [- 0.3 A^{3}]$

Step 3.2.2

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

$180 \cdot 1 + \frac{d}{d A} [- 0.3 A^{3}]$

Step 3.2.3

Multiply $180$ by $1$ .

$180 + \frac{d}{d A} [- 0.3 A^{3}]$

Step 3.3

Evaluate $\frac{d}{d A} [- 0.3 A^{3}]$ .

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Step 3.3.1

Since $- 0.3$ is constant with respect to $A$ , the derivative of $- 0.3 A^{3}$ with respect to $A$ is $- 0.3 \frac{d}{d A} [A^{3}]$ .

$180 - 0.3 \frac{d}{d A} [A^{3}]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d A} [A^{n}]$ is $n A^{n - 1}$ where $n = 3$ .

$180 - 0.3 (3 A^{2})$

Step 3.3.3

Multiply $3$ by $- 0.3$ .

$180 - 0.9 A^{2}$

Step 3.4

Reorder terms.

$- 0.9 A^{2} + 180$

Step 4

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

$0 = - 0.9 A^{2} + 180$

Step 5

Solve for $A$ .

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Step 5.1

Since $A$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 0.9 A^{2} + 180 = 0$

Step 5.2

Subtract $180$ from both sides of the equation.

$- 0.9 A^{2} = - 180$

Step 5.3

Divide each term in $- 0.9 A^{2} = - 180$ by $- 0.9$ and simplify.

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Step 5.3.1

Divide each term in $- 0.9 A^{2} = - 180$ by $- 0.9$ .

$\frac{- 0.9 A^{2}}{- 0.9} = \frac{- 180}{- 0.9}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 0.9$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 0.9 A^{2}}{- 0.9} = \frac{- 180}{- 0.9}$

Step 5.3.2.1.2

Divide $A^{2}$ by $1$ .

$A^{2} = \frac{- 180}{- 0.9}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $- 180$ by $- 0.9$ .

$A^{2} = 200$

Step 5.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$A = \pm \sqrt{200}$

Step 5.5

Simplify $\pm \sqrt{200}$ .

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Step 5.5.1

Rewrite $200$ as $10^{2} \cdot 2$ .

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Step 5.5.1.1

Factor $100$ out of $200$ .

$A = \pm \sqrt{100 (2)}$

Step 5.5.1.2

Rewrite $100$ as $10^{2}$ .

$A = \pm \sqrt{10^{2} \cdot 2}$

Step 5.5.2

Pull terms out from under the radical.

$A = \pm 10 \sqrt{2}$

Step 5.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.6.1

First, use the positive value of the $\pm$ to find the first solution.

$A = 10 \sqrt{2}$

Step 5.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$A = - 10 \sqrt{2}$

Step 5.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$A = 10 \sqrt{2}, - 10 \sqrt{2}$

Step 6

Replace $S'$ with $\frac{d S}{d A}$ .

$\frac{d S}{d A} = 10 \sqrt{2}, - 10 \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{d S}{d A} = 10 \sqrt{2}, - 10 \sqrt{2}$

Decimal Form:

$\frac{d S}{d A} = 14.14213562 \dots, - 14.14213562 \dots$