Calculus Examples

$y = 400 (15 - x^{2}) (3 x - 2)$

Step 1

Since $400$ is constant with respect to $x$ , the derivative of $400 (15 - x^{2}) (3 x - 2)$ with respect to $x$ is $400 \frac{d}{d x} [(15 - x^{2}) (3 x - 2)]$ .

$400 \frac{d}{d x} [(15 - x^{2}) (3 x - 2)]$

Step 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 $f (x) = 15 - x^{2}$ and $g (x) = 3 x - 2$ .

$400 ((15 - x^{2}) \frac{d}{d x} [3 x - 2] + (3 x - 2) \frac{d}{d x} [15 - x^{2}])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 x - 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$ .

$400 ((15 - x^{2}) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]) + (3 x - 2) \frac{d}{d x} [15 - x^{2}])$

Step 3.2

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

$400 ((15 - x^{2}) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + (3 x - 2) \frac{d}{d x} [15 - x^{2}])$

Step 3.3

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

$400 ((15 - x^{2}) (3 \cdot 1 + \frac{d}{d x} [- 2]) + (3 x - 2) \frac{d}{d x} [15 - x^{2}])$

Step 3.4

Multiply $3$ by $1$ .

$400 ((15 - x^{2}) (3 + \frac{d}{d x} [- 2]) + (3 x - 2) \frac{d}{d x} [15 - x^{2}])$

Step 3.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$400 ((15 - x^{2}) (3 + 0) + (3 x - 2) \frac{d}{d x} [15 - x^{2}])$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $3$ and $0$ .

$400 ((15 - x^{2}) \cdot 3 + (3 x - 2) \frac{d}{d x} [15 - x^{2}])$

Step 3.6.2

Move $3$ to the left of $15 - x^{2}$ .

$400 (3 \cdot (15 - x^{2}) + (3 x - 2) \frac{d}{d x} [15 - x^{2}])$

Step 3.7

By the Sum Rule, the derivative of $15 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [15] + \frac{d}{d x} [- x^{2}]$ .

$400 (3 (15 - x^{2}) + (3 x - 2) (\frac{d}{d x} [15] + \frac{d}{d x} [- x^{2}]))$

Step 3.8

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$400 (3 (15 - x^{2}) + (3 x - 2) (0 + \frac{d}{d x} [- x^{2}]))$

Step 3.9

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$400 (3 (15 - x^{2}) + (3 x - 2) \frac{d}{d x} [- x^{2}])$

Step 3.10

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

$400 (3 (15 - x^{2}) + (3 x - 2) (- \frac{d}{d x} [x^{2}]))$

Step 3.11

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

$400 (3 (15 - x^{2}) + (3 x - 2) (- (2 x)))$

Step 3.12

Multiply $2$ by $- 1$ .

$400 (3 (15 - x^{2}) + (3 x - 2) (- 2 x))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$400 (3 \cdot 15 + 3 (- x^{2}) + (3 x - 2) (- 2 x))$

Step 4.2

Apply the distributive property.

$400 (3 \cdot 15 + 3 (- x^{2}) + 3 x (- 2 x) - 2 (- 2 x))$

Step 4.3

Apply the distributive property.

$400 (3 \cdot 15) + 400 (3 (- x^{2})) + 400 (3 x (- 2 x)) + 400 (- 2 (- 2 x))$

Step 4.4

Combine terms.

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Step 4.4.1

Multiply $3$ by $15$ .

$400 \cdot 45 + 400 (3 (- x^{2})) + 400 (3 x (- 2 x)) + 400 (- 2 (- 2 x))$

Step 4.4.2

Multiply $400$ by $45$ .

$18000 + 400 (3 (- x^{2})) + 400 (3 x (- 2 x)) + 400 (- 2 (- 2 x))$

Step 4.4.3

Multiply $- 1$ by $3$ .

$18000 + 400 (- 3 x^{2}) + 400 (3 x (- 2 x)) + 400 (- 2 (- 2 x))$

Step 4.4.4

Multiply $- 3$ by $400$ .

$18000 - 1200 x^{2} + 400 (3 x (- 2 x)) + 400 (- 2 (- 2 x))$

Step 4.4.5

Multiply $- 2$ by $3$ .

$18000 - 1200 x^{2} + 400 (- 6 x \cdot x) + 400 (- 2 (- 2 x))$

Step 4.4.6

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

$18000 - 1200 x^{2} + 400 (- 6 (x^{1} x)) + 400 (- 2 (- 2 x))$

Step 4.4.7

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

$18000 - 1200 x^{2} + 400 (- 6 (x^{1} x^{1})) + 400 (- 2 (- 2 x))$

Step 4.4.8

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

$18000 - 1200 x^{2} + 400 (- 6 x^{1 + 1}) + 400 (- 2 (- 2 x))$

Step 4.4.9

Add $1$ and $1$ .

$18000 - 1200 x^{2} + 400 (- 6 x^{2}) + 400 (- 2 (- 2 x))$

Step 4.4.10

Multiply $- 6$ by $400$ .

$18000 - 1200 x^{2} - 2400 x^{2} + 400 (- 2 (- 2 x))$

Step 4.4.11

Multiply $- 2$ by $- 2$ .

$18000 - 1200 x^{2} - 2400 x^{2} + 400 (4 x)$

Step 4.4.12

Multiply $4$ by $400$ .

$18000 - 1200 x^{2} - 2400 x^{2} + 1600 x$

Step 4.4.13

Subtract $2400 x^{2}$ from $- 1200 x^{2}$ .

$18000 - 3600 x^{2} + 1600 x$

Step 4.5

Reorder terms.

$- 3600 x^{2} + 1600 x + 18000$