Calculus Examples

$f (x) = (5 - x) (9 - x)$

Step 1

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 $f (x) = 5 - x$ and $g (x) = 9 - x$ .

$(5 - x) \frac{d}{d x} [9 - x] + (9 - x) \frac{d}{d x} [5 - x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $9 - x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x]$ .

$(5 - x) (\frac{d}{d x} [9] + \frac{d}{d x} [- x]) + (9 - x) \frac{d}{d x} [5 - x]$

Step 2.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$(5 - x) (0 + \frac{d}{d x} [- x]) + (9 - x) \frac{d}{d x} [5 - x]$

Step 2.3

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

$(5 - x) \frac{d}{d x} [- x] + (9 - x) \frac{d}{d x} [5 - x]$

Step 2.4

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

$(5 - x) (- \frac{d}{d x} [x]) + (9 - x) \frac{d}{d x} [5 - x]$

Step 2.5

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

$(5 - x) (- 1 \cdot 1) + (9 - x) \frac{d}{d x} [5 - x]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $- 1$ by $1$ .

$(5 - x) \cdot - 1 + (9 - x) \frac{d}{d x} [5 - x]$

Step 2.6.2

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

$- 1 \cdot (5 - x) + (9 - x) \frac{d}{d x} [5 - x]$

Step 2.6.3

Rewrite $- 1 (5 - x)$ as $- (5 - x)$ .

$- (5 - x) + (9 - x) \frac{d}{d x} [5 - x]$

Step 2.7

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

$- (5 - x) + (9 - x) (\frac{d}{d x} [5] + \frac{d}{d x} [- x])$

Step 2.8

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

$- (5 - x) + (9 - x) (0 + \frac{d}{d x} [- x])$

Step 2.9

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

$- (5 - x) + (9 - x) \frac{d}{d x} [- x]$

Step 2.10

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

$- (5 - x) + (9 - x) (- \frac{d}{d x} [x])$

Step 2.11

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

$- (5 - x) + (9 - x) (- 1 \cdot 1)$

Step 2.12

Simplify the expression.

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Step 2.12.1

Multiply $- 1$ by $1$ .

$- (5 - x) + (9 - x) \cdot - 1$

Step 2.12.2

Move $- 1$ to the left of $9 - x$ .

$- (5 - x) - 1 \cdot (9 - x)$

Step 2.12.3

Rewrite $- 1 (9 - x)$ as $- (9 - x)$ .

$- (5 - x) - (9 - x)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$- 1 \cdot 5 - - x - (9 - x)$

Step 3.2

Apply the distributive property.

$- 1 \cdot 5 - - x - 1 \cdot 9 - - x$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $- 1$ by $5$ .

$- 5 - - x - 1 \cdot 9 - - x$

Step 3.3.2

Multiply $- 1$ by $- 1$ .

$- 5 + 1 x - 1 \cdot 9 - - x$

Step 3.3.3

Multiply $x$ by $1$ .

$- 5 + x - 1 \cdot 9 - - x$

Step 3.3.4

Multiply $- 1$ by $9$ .

$- 5 + x - 9 - - x$

Step 3.3.5

Multiply $- 1$ by $- 1$ .

$- 5 + x - 9 + 1 x$

Step 3.3.6

Multiply $x$ by $1$ .

$- 5 + x - 9 + x$

Step 3.3.7

Subtract $9$ from $- 5$ .

$x - 14 + x$

Step 3.3.8

Add $x$ and $x$ .

$2 x - 14$