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

The derivative of $s$ with respect to $A$ is $s'$ .

$s'$

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.

$s' = - 0.9 A^{2} + 180$

Step 5

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

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