Calculus Examples

$f (x) = 10 (3 x + 1) (1 - 5 x)$

Step 1

Since $10$ is constant with respect to $x$ , the derivative of $10 (3 x + 1) (1 - 5 x)$ with respect to $x$ is $10 \frac{d}{d x} [(3 x + 1) (1 - 5 x)]$ .

$10 \frac{d}{d x} [(3 x + 1) (1 - 5 x)]$

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 $f (x) = 3 x + 1$ and $g (x) = 1 - 5 x$ .

$10 ((3 x + 1) \frac{d}{d x} [1 - 5 x] + (1 - 5 x) \frac{d}{d x} [3 x + 1])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 - 5 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 5 x]$ .

$10 ((3 x + 1) (\frac{d}{d x} [1] + \frac{d}{d x} [- 5 x]) + (1 - 5 x) \frac{d}{d x} [3 x + 1])$

Step 3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$10 ((3 x + 1) (0 + \frac{d}{d x} [- 5 x]) + (1 - 5 x) \frac{d}{d x} [3 x + 1])$

Step 3.3

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

$10 ((3 x + 1) \frac{d}{d x} [- 5 x] + (1 - 5 x) \frac{d}{d x} [3 x + 1])$

Step 3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$10 ((3 x + 1) (- 5 \frac{d}{d x} [x]) + (1 - 5 x) \frac{d}{d x} [3 x + 1])$

Step 3.5

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

$10 ((3 x + 1) (- 5 \cdot 1) + (1 - 5 x) \frac{d}{d x} [3 x + 1])$

Step 3.6

Simplify the expression.

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Step 3.6.1

Multiply $- 5$ by $1$ .

$10 ((3 x + 1) \cdot - 5 + (1 - 5 x) \frac{d}{d x} [3 x + 1])$

Step 3.6.2

Move $- 5$ to the left of $3 x + 1$ .

$10 (- 5 \cdot (3 x + 1) + (1 - 5 x) \frac{d}{d x} [3 x + 1])$

Step 3.7

By the Sum Rule, the derivative of $3 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [1]$ .

$10 (- 5 (3 x + 1) + (1 - 5 x) (\frac{d}{d x} [3 x] + \frac{d}{d x} [1]))$

Step 3.8

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

$10 (- 5 (3 x + 1) + (1 - 5 x) (3 \frac{d}{d x} [x] + \frac{d}{d x} [1]))$

Step 3.9

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

$10 (- 5 (3 x + 1) + (1 - 5 x) (3 \cdot 1 + \frac{d}{d x} [1]))$

Step 3.10

Multiply $3$ by $1$ .

$10 (- 5 (3 x + 1) + (1 - 5 x) (3 + \frac{d}{d x} [1]))$

Step 3.11

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$10 (- 5 (3 x + 1) + (1 - 5 x) (3 + 0))$

Step 3.12

Simplify the expression.

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Step 3.12.1

Add $3$ and $0$ .

$10 (- 5 (3 x + 1) + (1 - 5 x) \cdot 3)$

Step 3.12.2

Move $3$ to the left of $1 - 5 x$ .

$10 (- 5 (3 x + 1) + 3 \cdot (1 - 5 x))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$10 (- 5 (3 x) - 5 \cdot 1 + 3 (1 - 5 x))$

Step 4.2

Apply the distributive property.

$10 (- 5 (3 x) - 5 \cdot 1 + 3 \cdot 1 + 3 (- 5 x))$

Step 4.3

Apply the distributive property.

$10 (- 5 (3 x)) + 10 (- 5 \cdot 1) + 10 (3 \cdot 1) + 10 (3 (- 5 x))$

Step 4.4

Combine terms.

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Step 4.4.1

Multiply $3$ by $- 5$ .

$10 (- 15 x) + 10 (- 5 \cdot 1) + 10 (3 \cdot 1) + 10 (3 (- 5 x))$

Step 4.4.2

Multiply $- 15$ by $10$ .

$- 150 x + 10 (- 5 \cdot 1) + 10 (3 \cdot 1) + 10 (3 (- 5 x))$

Step 4.4.3

Multiply $- 5$ by $1$ .

$- 150 x + 10 \cdot - 5 + 10 (3 \cdot 1) + 10 (3 (- 5 x))$

Step 4.4.4

Multiply $10$ by $- 5$ .

$- 150 x - 50 + 10 (3 \cdot 1) + 10 (3 (- 5 x))$

Step 4.4.5

Multiply $3$ by $1$ .

$- 150 x - 50 + 10 \cdot 3 + 10 (3 (- 5 x))$

Step 4.4.6

Multiply $10$ by $3$ .

$- 150 x - 50 + 30 + 10 (3 (- 5 x))$

Step 4.4.7

Multiply $- 5$ by $3$ .

$- 150 x - 50 + 30 + 10 (- 15 x)$

Step 4.4.8

Multiply $- 15$ by $10$ .

$- 150 x - 50 + 30 - 150 x$

Step 4.4.9

Add $- 50$ and $30$ .

$- 150 x - 20 - 150 x$

Step 4.4.10

Subtract $150 x$ from $- 150 x$ .

$- 300 x - 20$