Calculus Examples

$f (x) = 100 x (2 x + 3) (x - 5)$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $100$ is constant with respect to $x$ , the derivative of $100 x (2 x + 3) (x - 5)$ with respect to $x$ is $100 \frac{d}{d x} [(x (2 x + 3)) (x - 5)]$ .

$100 \frac{d}{d x} [(x (2 x + 3)) (x - 5)]$

Step 1.1.2

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 $f (x) = x (2 x + 3)$ and $g (x) = x - 5$ .

$100 (x (2 x + 3) \frac{d}{d x} [x - 5] + (x - 5) \frac{d}{d x} [x (2 x + 3)])$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$100 (x (2 x + 3) (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]) + (x - 5) \frac{d}{d x} [x (2 x + 3)])$

Step 1.1.3.2

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

$100 (x (2 x + 3) (1 + \frac{d}{d x} [- 5]) + (x - 5) \frac{d}{d x} [x (2 x + 3)])$

Step 1.1.3.3

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$100 (x (2 x + 3) (1 + 0) + (x - 5) \frac{d}{d x} [x (2 x + 3)])$

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Add $1$ and $0$ .

$100 (x (2 x + 3) \cdot 1 + (x - 5) \frac{d}{d x} [x (2 x + 3)])$

Step 1.1.3.4.2

Multiply $x$ by $1$ .

$100 (x (2 x + 3) + (x - 5) \frac{d}{d x} [x (2 x + 3)])$

Step 1.1.4

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 $g (x) = 2 x + 3$ .

$100 (x (2 x + 3) + (x - 5) (x \frac{d}{d x} [2 x + 3] + (2 x + 3) \frac{d}{d x} [x]))$

Step 1.1.5

Differentiate.

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Step 1.1.5.1

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

$100 (x (2 x + 3) + (x - 5) (x (\frac{d}{d x} [2 x] + \frac{d}{d x} [3]) + (2 x + 3) \frac{d}{d x} [x]))$

Step 1.1.5.2

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

$100 (x (2 x + 3) + (x - 5) (x (2 \frac{d}{d x} [x] + \frac{d}{d x} [3]) + (2 x + 3) \frac{d}{d x} [x]))$

Step 1.1.5.3

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

$100 (x (2 x + 3) + (x - 5) (x (2 \cdot 1 + \frac{d}{d x} [3]) + (2 x + 3) \frac{d}{d x} [x]))$

Step 1.1.5.4

Multiply $2$ by $1$ .

$100 (x (2 x + 3) + (x - 5) (x (2 + \frac{d}{d x} [3]) + (2 x + 3) \frac{d}{d x} [x]))$

Step 1.1.5.5

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$100 (x (2 x + 3) + (x - 5) (x (2 + 0) + (2 x + 3) \frac{d}{d x} [x]))$

Step 1.1.5.6

Simplify the expression.

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Step 1.1.5.6.1

Add $2$ and $0$ .

$100 (x (2 x + 3) + (x - 5) (x \cdot 2 + (2 x + 3) \frac{d}{d x} [x]))$

Step 1.1.5.6.2

Move $2$ to the left of $x$ .

$100 (x (2 x + 3) + (x - 5) (2 \cdot x + (2 x + 3) \frac{d}{d x} [x]))$

Step 1.1.5.7

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

$100 (x (2 x + 3) + (x - 5) (2 x + (2 x + 3) \cdot 1))$

Step 1.1.5.8

Simplify by adding terms.

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Step 1.1.5.8.1

Multiply $2 x + 3$ by $1$ .

$100 (x (2 x + 3) + (x - 5) (2 x + 2 x + 3))$

Step 1.1.5.8.2

Add $2 x$ and $2 x$ .

$100 (x (2 x + 3) + (x - 5) (4 x + 3))$

Step 1.1.6

Simplify.

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Step 1.1.6.1