Calculus Examples

$s = 330 A - 0.2 A^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d A} (s) = \frac{d}{d A} (330 A - 0.2 A^{3})$

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 $330 A - 0.2 A^{3}$ with respect to $A$ is $\frac{d}{d A} [330 A] + \frac{d}{d A} [- 0.2 A^{3}]$ .

$\frac{d}{d A} [330 A] + \frac{d}{d A} [- 0.2 A^{3}]$

Step 3.2

Evaluate $\frac{d}{d A} [330 A]$ .

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Step 3.2.1

Since $330$ is constant with respect to $A$ , the derivative of $330 A$ with respect to $A$ is $330 \frac{d}{d A} [A]$ .

$330 \frac{d}{d A} [A] + \frac{d}{d A} [- 0.2 A^{3}]$

Step 3.2.2

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

$330 \cdot 1 + \frac{d}{d A} [- 0.2 A^{3}]$

Step 3.2.3

Multiply $330$ by $1$ .

$330 + \frac{d}{d A} [- 0.2 A^{3}]$

Step 3.3

Evaluate $\frac{d}{d A} [- 0.2 A^{3}]$ .

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Step 3.3.1

Since $- 0.2$ is constant with respect to $A$ , the derivative of $- 0.2 A^{3}$ with respect to $A$ is $- 0.2 \frac{d}{d A} [A^{3}]$ .

$330 - 0.2 \frac{d}{d A} [A^{3}]$

Step 3.3.2

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

$330 - 0.2 (3 A^{2})$

Step 3.3.3

Multiply $3$ by $- 0.2$ .

$330 - 0.6 A^{2}$

Step 3.4

Reorder terms.

$- 0.6 A^{2} + 330$

Step 4

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

$0 = - 0.6 A^{2} + 330$

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.

$- 0.6 A^{2} + 330 = 0$

Step 5.2

Subtract $330$ from both sides of the equation.

$- 0.6 A^{2} = - 330$

Step 5.3

Divide each term in $- 0.6 A^{2} = - 330$ by $- 0.6$ and simplify.

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Step 5.3.1

Divide each term in $- 0.6 A^{2} = - 330$ by $- 0.6$ .

$\frac{- 0.6 A^{2}}{- 0.6} = \frac{- 330}{- 0.6}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 0.6$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 0.6 A^{2}}{- 0.6} = \frac{- 330}{- 0.6}$

Step 5.3.2.1.2

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

$A^{2} = \frac{- 330}{- 0.6}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $- 330$ by $- 0.6$ .

$A^{2} = 550$

Step 5.4

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

$A = \pm \sqrt{550}$

Step 5.5

Simplify $\pm \sqrt{550}$ .

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Step 5.5.1

Rewrite $550$ as $5^{2} \cdot 22$ .

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Step 5.5.1.1

Factor $25$ out of $550$ .

$A = \pm \sqrt{25 (22)}$

Step 5.5.1.2

Rewrite $25$ as $5^{2}$ .

$A = \pm \sqrt{5^{2} \cdot 22}$

Step 5.5.2

Pull terms out from under the radical.

$A = \pm 5 \sqrt{22}$

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.

$A = 5 \sqrt{22}$

Step 5.6.2

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

$A = - 5 \sqrt{22}$

Step 5.6.3

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

$A = 5 \sqrt{22}, - 5 \sqrt{22}$

Step 6

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

$\frac{d S}{d A} = 5 \sqrt{22}, - 5 \sqrt{22}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{d S}{d A} = 5 \sqrt{22}, - 5 \sqrt{22}$

Decimal Form:

$\frac{d S}{d A} = 23.45207879 \dots, - 23.45207879 \dots$