Calculus Examples

$y = (4 - x) (3 + x)$

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 $f (x) = 4 - x$ and $g (x) = 3 + x$ .

$(4 - x) \frac{d}{d x} [3 + x] + (3 + x) \frac{d}{d x} [4 - x]$

Step 2

Differentiate.

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Step 2.1

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

$(4 - x) (\frac{d}{d x} [3] + \frac{d}{d x} [x]) + (3 + x) \frac{d}{d x} [4 - x]$

Step 2.2

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

$(4 - x) (0 + \frac{d}{d x} [x]) + (3 + x) \frac{d}{d x} [4 - x]$

Step 2.3

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

$(4 - x) \frac{d}{d x} [x] + (3 + x) \frac{d}{d x} [4 - x]$

Step 2.4

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

$(4 - x) \cdot 1 + (3 + x) \frac{d}{d x} [4 - x]$

Step 2.5

Multiply $4 - x$ by $1$ .

$4 - x + (3 + x) \frac{d}{d x} [4 - x]$

Step 2.6

By the Sum Rule, the derivative of $4 - x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x]$ .

$4 - x + (3 + x) (\frac{d}{d x} [4] + \frac{d}{d x} [- x])$

Step 2.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$4 - x + (3 + x) (0 + \frac{d}{d x} [- x])$

Step 2.8

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

$4 - x + (3 + x) \frac{d}{d x} [- x]$

Step 2.9

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

$4 - x + (3 + x) (- \frac{d}{d x} [x])$

Step 2.10

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

$4 - x + (3 + x) (- 1 \cdot 1)$

Step 2.11

Simplify the expression.

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Step 2.11.1

Multiply $- 1$ by $1$ .

$4 - x + (3 + x) \cdot - 1$

Step 2.11.2

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

$4 - x - 1 \cdot (3 + x)$

Step 2.11.3

Rewrite $- 1 (3 + x)$ as $- (3 + x)$ .

$4 - x - (3 + x)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$4 - x - 1 \cdot 3 - x$

Step 3.2

Combine terms.

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Step 3.2.1

Multiply $- 1$ by $3$ .

$4 - x - 3 - x$

Step 3.2.2

Subtract $3$ from $4$ .

$- x + 1 - x$

Step 3.2.3

Subtract $x$ from $- x$ .

$- 2 x + 1$