Calculus Examples

$R (x) = (40000 + \frac{10000}{4} x) (24 - x) + (40000 + \frac{10000}{4} x) (4)$

Step 1

Differentiate using the Sum Rule.

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Step 1.1

Divide $10000$ by $4$ .

$\frac{d}{d x} [(40000 + 2500 x) (24 - x) + (40000 + \frac{10000}{4} x) (4)]$

Step 1.2

Divide $10000$ by $4$ .

$\frac{d}{d x} [(40000 + 2500 x) (24 - x) + (40000 + 2500 x) \cdot 4]$

Step 1.3

By the Sum Rule, the derivative of $(40000 + 2500 x) (24 - x) + (40000 + 2500 x) \cdot 4$ with respect to $x$ is $\frac{d}{d x} [(40000 + 2500 x) (24 - x)] + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$ .

$\frac{d}{d x} [(40000 + 2500 x) (24 - x)] + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2

Evaluate $\frac{d}{d x} [(40000 + 2500 x) (24 - x)]$ .

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 40000 + 2500 x$ and $g (x) = 24 - x$ .

$(40000 + 2500 x) \frac{d}{d x} [24 - x] + (24 - x) \frac{d}{d x} [40000 + 2500 x] + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.2

By the Sum Rule, the derivative of $24 - x$ with respect to $x$ is $\frac{d}{d x} [24] + \frac{d}{d x} [- x]$ .

$(40000 + 2500 x) (\frac{d}{d x} [24] + \frac{d}{d x} [- x]) + (24 - x) \frac{d}{d x} [40000 + 2500 x] + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.3

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$(40000 + 2500 x) (0 + \frac{d}{d x} [- x]) + (24 - x) \frac{d}{d x} [40000 + 2500 x] + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$(40000 + 2500 x) (0 - \frac{d}{d x} [x]) + (24 - x) \frac{d}{d x} [40000 + 2500 x] + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.5

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

$(40000 + 2500 x) (0 - 1 \cdot 1) + (24 - x) \frac{d}{d x} [40000 + 2500 x] + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.6

By the Sum Rule, the derivative of $40000 + 2500 x$ with respect to $x$ is $\frac{d}{d x} [40000] + \frac{d}{d x} [2500 x]$ .

$(40000 + 2500 x) (0 - 1 \cdot 1) + (24 - x) (\frac{d}{d x} [40000] + \frac{d}{d x} [2500 x]) + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.7

Since $40000$ is constant with respect to $x$ , the derivative of $40000$ with respect to $x$ is $0$ .

$(40000 + 2500 x) (0 - 1 \cdot 1) + (24 - x) (0 + \frac{d}{d x} [2500 x]) + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.8

Since $2500$ is constant with respect to $x$ , the derivative of $2500 x$ with respect to $x$ is $2500 \frac{d}{d x} [x]$ .

$(40000 + 2500 x) (0 - 1 \cdot 1) + (24 - x) (0 + 2500 \frac{d}{d x} [x]) + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.9

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

$(40000 + 2500 x) (0 - 1 \cdot 1) + (24 - x) (0 + 2500 \cdot 1) + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.10

Multiply $- 1$ by $1$ .

$(40000 + 2500 x) (0 - 1) + (24 - x) (0 + 2500 \cdot 1) + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.11

Subtract $1$ from $0$ .

$(40000 + 2500 x) \cdot - 1 + (24 - x) (0 + 2500 \cdot 1) + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.12

Move $- 1$ to the left of $40000 + 2500 x$ .

$- 1 \cdot (40000 + 2500 x) + (24 - x) (0 + 2500 \cdot 1) + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.13

Rewrite $- 1 (40000 + 2500 x)$ as $- (40000 + 2500 x)$ .

$- (40000 + 2500 x) + (24 - x) (0 + 2500 \cdot 1) + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.14

Multiply $2500$ by $1$ .

$- (40000 + 2500 x) + (24 - x) (0 + 2500) + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.15

Add $0$ and $2500$ .

$- (40000 + 2500 x) + (24 - x) \cdot 2500 + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 2.16

Move $2500$ to the left of $24 - x$ .

$- (40000 + 2500 x) + 2500 (24 - x) + \frac{d}{d x} [(40000 + 2500 x) \cdot 4]$

Step 3

Evaluate $\frac{d}{d x} [(40000 + 2500 x) \cdot 4]$ .

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Step 3.1

Move $4$ to the left of $40000 + 2500 x$ .

$- (40000 + 2500 x) + 2500 (24 - x) + \frac{d}{d x} [4 \cdot (40000 + 2500 x)]$

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4 (40000 + 2500 x)$ with respect to $x$ is $4 \frac{d}{d x} [40000 + 2500 x]$ .

$- (40000 + 2500 x) + 2500 (24 - x) + 4 \frac{d}{d x} [40000 + 2500 x]$

Step 3.3

By the Sum Rule, the derivative of $40000 + 2500 x$ with respect to $x$ is $\frac{d}{d x} [40000] + \frac{d}{d x} [2500 x]$ .

$- (40000 + 2500 x) + 2500 (24 - x) + 4 (\frac{d}{d x} [40000] + \frac{d}{d x} [2500 x])$

Step 3.4

Since $40000$ is constant with respect to $x$ , the derivative of $40000$ with respect to $x$ is $0$ .

$- (40000 + 2500 x) + 2500 (24 - x) + 4 (0 + \frac{d}{d x} [2500 x])$

Step 3.5

Since $2500$ is constant with respect to $x$ , the derivative of $2500 x$ with respect to $x$ is $2500 \frac{d}{d x} [x]$ .

$- (40000 + 2500 x) + 2500 (24 - x) + 4 (0 + 2500 \frac{d}{d x} [x])$

Step 3.6

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

$- (40000 + 2500 x) + 2500 (24 - x) + 4 (0 + 2500 \cdot 1)$

Step 3.7

Multiply $2500$ by $1$ .

$- (40000 + 2500 x) + 2500 (24 - x) + 4 (0 + 2500)$

Step 3.8

Add $0$ and $2500$ .

$- (40000 + 2500 x) + 2500 (24 - x) + 4 \cdot 2500$

Step 3.9

Multiply $4$ by $2500$ .

$- (40000 + 2500 x) + 2500 (24 - x) + 10000$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$- 1 \cdot 40000 - (2500 x) + 2500 (24 - x) + 10000$

Step 4.2

Apply the distributive property.

$- 1 \cdot 40000 - (2500 x) + 2500 \cdot 24 + 2500 (- x) + 10000$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $- 1$ by $40000$ .

$- 40000 - (2500 x) + 2500 \cdot 24 + 2500 (- x) + 10000$

Step 4.3.2

Multiply $2500$ by $- 1$ .

$- 40000 - 2500 x + 2500 \cdot 24 + 2500 (- x) + 10000$

Step 4.3.3

Multiply $2500$ by $24$ .

$- 40000 - 2500 x + 60000 + 2500 (- x) + 10000$

Step 4.3.4

Multiply $- 1$ by $2500$ .

$- 40000 - 2500 x + 60000 - 2500 x + 10000$

Step 4.3.5

Add $- 40000$ and $60000$ .

$- 2500 x + 20000 - 2500 x + 10000$

Step 4.3.6

Subtract $2500 x$ from $- 2500 x$ .

$- 5000 x + 20000 + 10000$

Step 4.3.7

Add $20000$ and $10000$ .

$- 5000 x + 30000$