Calculus Examples

$100 - 1 (x - 70) (70 + x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $100 - 1 (x - 70) (70 + x)$ with respect to $x$ is $\frac{d}{d x} [100] + \frac{d}{d x} [- 1 (x - 70) (70 + x)]$ .

$\frac{d}{d x} [100] + \frac{d}{d x} [- 1 (x - 70) (70 + x)]$

Step 1.2

Since $100$ is constant with respect to $x$ , the derivative of $100$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 1 (x - 70) (70 + x)]$

Step 2

Evaluate $\frac{d}{d x} [- 1 (x - 70) (70 + x)]$ .

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Step 2.1

Rewrite $- 1 (x - 70)$ as $- (x - 70)$ .

$0 + \frac{d}{d x} [- (x - 70) (70 + x)]$

Step 2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- (x - 70) (70 + x)$ with respect to $x$ is $- \frac{d}{d x} [(x - 70) (70 + x)]$ .

$0 - \frac{d}{d x} [(x - 70) (70 + x)]$

Step 2.3

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 $f (x) = x - 70$ and $g (x) = 70 + x$ .

$0 - ((x - 70) \frac{d}{d x} [70 + x] + (70 + x) \frac{d}{d x} [x - 70])$

Step 2.4

By the Sum Rule, the derivative of $70 + x$ with respect to $x$ is $\frac{d}{d x} [70] + \frac{d}{d x} [x]$ .

$0 - ((x - 70) (\frac{d}{d x} [70] + \frac{d}{d x} [x]) + (70 + x) \frac{d}{d x} [x - 70])$

Step 2.5

Since $70$ is constant with respect to $x$ , the derivative of $70$ with respect to $x$ is $0$ .

$0 - ((x - 70) (0 + \frac{d}{d x} [x]) + (70 + x) \frac{d}{d x} [x - 70])$

Step 2.6

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

$0 - ((x - 70) (0 + 1) + (70 + x) \frac{d}{d x} [x - 70])$

Step 2.7

By the Sum Rule, the derivative of $x - 70$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 70]$ .

$0 - ((x - 70) (0 + 1) + (70 + x) (\frac{d}{d x} [x] + \frac{d}{d x} [- 70]))$

Step 2.8

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

$0 - ((x - 70) (0 + 1) + (70 + x) (1 + \frac{d}{d x} [- 70]))$

Step 2.9

Since $- 70$ is constant with respect to $x$ , the derivative of $- 70$ with respect to $x$ is $0$ .

$0 - ((x - 70) (0 + 1) + (70 + x) (1 + 0))$

Step 2.10

Add $0$ and $1$ .

$0 - ((x - 70) \cdot 1 + (70 + x) (1 + 0))$

Step 2.11

Multiply $x - 70$ by $1$ .

$0 - (x - 70 + (70 + x) (1 + 0))$

Step 2.12

Add $1$ and $0$ .

$0 - (x - 70 + (70 + x) \cdot 1)$

Step 2.13

Multiply $70 + x$ by $1$ .

$0 - (x - 70 + 70 + x)$

Step 2.14

Add $- 70$ and $70$ .

$0 - (x + 0 + x)$

Step 2.15

Add $x$ and $0$ .

$0 - (x + x)$

Step 2.16

Add $x$ and $x$ .

$0 - (2 x)$

Step 2.17

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 3

Subtract $2 x$ from $0$ .

$- 2 x$