Calculus Examples

$y = 7 x (3 - x)$ , $x = 3$

Step 1

Since $7$ is constant with respect to $x$ , the derivative of $7 x (3 - x)$ with respect to $x$ is $7 \frac{d}{d x} [x (3 - x)]$ .

$7 \frac{d}{d x} [x (3 - x)]$

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

$7 (x \frac{d}{d x} [3 - x] + (3 - x) \frac{d}{d x} [x])$

Step 3

Differentiate.

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Step 3.1

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

$7 (x (\frac{d}{d x} [3] + \frac{d}{d x} [- x]) + (3 - x) \frac{d}{d x} [x])$

Step 3.2

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

$7 (x (0 + \frac{d}{d x} [- x]) + (3 - x) \frac{d}{d x} [x])$

Step 3.3

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

$7 (x \frac{d}{d x} [- x] + (3 - x) \frac{d}{d x} [x])$

Step 3.4

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

$7 (x (- \frac{d}{d x} [x]) + (3 - x) \frac{d}{d x} [x])$

Step 3.5

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

$7 (x (- 1 \cdot 1) + (3 - x) \frac{d}{d x} [x])$

Step 3.6

Simplify the expression.

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Step 3.6.1

Multiply $- 1$ by $1$ .

$7 (x \cdot - 1 + (3 - x) \frac{d}{d x} [x])$

Step 3.6.2

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

$7 (- 1 \cdot x + (3 - x) \frac{d}{d x} [x])$

Step 3.6.3

Rewrite $- 1 x$ as $- x$ .

$7 (- x + (3 - x) \frac{d}{d x} [x])$

Step 3.7

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

$7 (- x + (3 - x) \cdot 1)$

Step 3.8

Simplify by adding terms.

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Step 3.8.1

Multiply $3 - x$ by $1$ .

$7 (- x + 3 - x)$

Step 3.8.2

Subtract $x$ from $- x$ .

$7 (- 2 x + 3)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$7 (- 2 x) + 7 \cdot 3$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $- 2$ by $7$ .

$- 14 x + 7 \cdot 3$

Step 4.2.2

Multiply $7$ by $3$ .

$- 14 x + 21$

Step 5

Evaluate the derivative at $x = 3$ .

$- 14 \cdot 3 + 21$

Step 6

Simplify.

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Step 6.1

Multiply $- 14$ by $3$ .

$- 42 + 21$

Step 6.2

Add $- 42$ and $21$ .

$- 21$

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