Calculus Examples

$R = 2 x (900 + 32 x - x^{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (R) = \frac{d}{d x} (2 x (900 + 32 x - x^{2}))$

Step 2

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

$R'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x (900 + 32 x - x^{2})$ with respect to $x$ is $2 \frac{d}{d x} [x (900 + 32 x - x^{2})]$ .

$2 \frac{d}{d x} [x (900 + 32 x - x^{2})]$

Step 3.2

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 $f (x) = x$ and $g (x) = 900 + 32 x - x^{2}$ .

$2 (x \frac{d}{d x} [900 + 32 x - x^{2}] + (900 + 32 x - x^{2}) \frac{d}{d x} [x])$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $900 + 32 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [900] + \frac{d}{d x} [32 x] + \frac{d}{d x} [- x^{2}]$ .

$2 (x (\frac{d}{d x} [900] + \frac{d}{d x} [32 x] + \frac{d}{d x} [- x^{2}]) + (900 + 32 x - x^{2}) \frac{d}{d x} [x])$

Step 3.3.2

Since $900$ is constant with respect to $x$ , the derivative of $900$ with respect to $x$ is $0$ .

$2 (x (0 + \frac{d}{d x} [32 x] + \frac{d}{d x} [- x^{2}]) + (900 + 32 x - x^{2}) \frac{d}{d x} [x])$

Step 3.3.3

Add $0$ and $\frac{d}{d x} [32 x]$ .

$2 (x (\frac{d}{d x} [32 x] + \frac{d}{d x} [- x^{2}]) + (900 + 32 x - x^{2}) \frac{d}{d x} [x])$

Step 3.3.4

Since $32$ is constant with respect to $x$ , the derivative of $32 x$ with respect to $x$ is $32 \frac{d}{d x} [x]$ .

$2 (x (32 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}]) + (900 + 32 x - x^{2}) \frac{d}{d x} [x])$

Step 3.3.5

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

$2 (x (32 \cdot 1 + \frac{d}{d x} [- x^{2}]) + (900 + 32 x - x^{2}) \frac{d}{d x} [x])$

Step 3.3.6

Multiply $32$ by $1$ .

$2 (x (32 + \frac{d}{d x} [- x^{2}]) + (900 + 32 x - x^{2}) \frac{d}{d x} [x])$

Step 3.3.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$2 (x (32 - \frac{d}{d x} [x^{2}]) + (900 + 32 x - x^{2}) \frac{d}{d x} [x])$

Step 3.3.8

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

$2 (x (32 - (2 x)) + (900 + 32 x - x^{2}) \frac{d}{d x} [x])$

Step 3.3.9

Multiply $2$ by $- 1$ .

$2 (x (32 - 2 x) + (900 + 32 x - x^{2}) \frac{d}{d x} [x])$

Step 3.3.10

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

$2 (x (32 - 2 x) + (900 + 32 x - x^{2}) \cdot 1)$

Step 3.3.11

Multiply $900 + 32 x - x^{2}$ by $1$ .

$2 (x (32 - 2 x) + 900 + 32 x - x^{2})$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$2 (x \cdot 32 + x (- 2 x) + 900 + 32 x - x^{2})$

Step 3.4.2

Apply the distributive property.

$2 (x \cdot 32) + 2 (x (- 2 x)) + 2 \cdot 900 + 2 (32 x) + 2 (- x^{2})$

Step 3.4.3

Combine terms.

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Step 3.4.3.1

Move $32$ to the left of $x$ .

$2 (32 \cdot x) + 2 (x (- 2 x)) + 2 \cdot 900 + 2 (32 x) + 2 (- x^{2})$

Step 3.4.3.2

Multiply $32$ by $2$ .

$64 x + 2 (x (- 2 x)) + 2 \cdot 900 + 2 (32 x) + 2 (- x^{2})$

Step 3.4.3.3

Raise $x$ to the power of $1$ .

$64 x + 2 (- 2 (x^{1} x)) + 2 \cdot 900 + 2 (32 x) + 2 (- x^{2})$

Step 3.4.3.4

Raise $x$ to the power of $1$ .

$64 x + 2 (- 2 (x^{1} x^{1})) + 2 \cdot 900 + 2 (32 x) + 2 (- x^{2})$

Step 3.4.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$64 x + 2 (- 2 x^{1 + 1}) + 2 \cdot 900 + 2 (32 x) + 2 (- x^{2})$

Step 3.4.3.6

Add $1$ and $1$ .

$64 x + 2 (- 2 x^{2}) + 2 \cdot 900 + 2 (32 x) + 2 (- x^{2})$

Step 3.4.3.7

Multiply $- 2$ by $2$ .

$64 x - 4 x^{2} + 2 \cdot 900 + 2 (32 x) + 2 (- x^{2})$

Step 3.4.3.8

Multiply $2$ by $900$ .

$64 x - 4 x^{2} + 1800 + 2 (32 x) + 2 (- x^{2})$

Step 3.4.3.9

Multiply $32$ by $2$ .

$64 x - 4 x^{2} + 1800 + 64 x + 2 (- x^{2})$

Step 3.4.3.10

Add $64 x$ and $64 x$ .

$128 x - 4 x^{2} + 1800 + 2 (- x^{2})$

Step 3.4.3.11

Multiply $- 1$ by $2$ .

$128 x - 4 x^{2} + 1800 - 2 x^{2}$

Step 3.4.3.12

Subtract $2 x^{2}$ from $- 4 x^{2}$ .

$128 x - 6 x^{2} + 1800$

Step 3.4.4

Reorder terms.

$- 6 x^{2} + 128 x + 1800$

Step 4

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

$R' = - 6 x^{2} + 128 x + 1800$

Step 5

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

$\frac{d R}{d x} = - 6 x^{2} + 128 x + 1800$