## Differentiation of a Product

A Differentiation of a Product calculator automates the process of finding the derivative of the product of two functions. By inputting the two functions, the calculator applies the product rule \((uv)’ = u \cdot v’ + v \cdot u’\) and provides the derivative quickly and accurately, saving time and reducing errors.

### Differentiation of a Product Using the Product Rule

#### Introduction

Differentiation is a fundamental concept in calculus used to determine the rate at which a function changes. When dealing with the product of two functions, the product rule provides an efficient method for finding the derivative.

#### Product Rule Definition

The product rule states that if you have two differentiable functions, \( u(x) \) and \( v(x) \), their product \( y = u(x) \cdot v(x) \) is differentiable, and its derivative is given by:

\[

\frac{d}{dx}[u(x) \cdot v(x)] = u(x) \cdot \frac{d}{dx}v(x) + v(x) \cdot \frac{d}{dx}u(x)

\]

In more concise notation:

\[

(y)’ = u \cdot v’ + v \cdot u’

\]

#### Step-by-Step Process

1. Identify the two functions:

– Recognize the two functions \( u(x) \) and \( v(x) \) that are being multiplied.

2. Differentiate each function separately:

– Find the derivative of \( u(x) \), denoted as \( u'(x) \) or \( \frac{du}{dx} \).

– Find the derivative of \( v(x) \), denoted as \( v'(x) \) or \( \frac{dv}{dx} \).

3. Apply the product rule:

– Substitute \( u(x) \), \( v(x) \), \( u'(x) \), and \( v'(x) \) into the product rule formula:

\[

(uv)’ = u \cdot v’ + v \cdot u’

\]

4. Simplify the expression:

– Combine and simplify the terms to obtain the final derivative.

#### Example

Let’s differentiate \( y = (3x^2) \cdot (2x) \) using the product rule.

Step 1: Identify the two functions:

– \( u(x) = 3x^2 \)

– \( v(x) = 2x \)

Step 2: Differentiate each function separately:

– \( u'(x) = \frac{d}{dx}[3x^2] = 6x \)

– \( v'(x) = \frac{d}{dx}[2x] = 2 \)

Step 3: Apply the product rule:

\[

(y)’ = u \cdot v’ + v \cdot u’

\]

\[

(y)’ = (3x^2) \cdot (2) + (2x) \cdot (6x)

\]

Step 4: Simplify the expression:

\[

(y)’ = 6x^2 + 12x^2

\]

\[

(y)’ = 18x^2

\]

So, the derivative of \( y = (3x^2) \cdot (2x) \) is \( 18x^2 \).

#### Vital Tips for Better Understanding

1. Practice basic differentiation:

– Ensure you are comfortable with finding the derivatives of basic functions before applying the product rule.

2. Memorize the product rule:

– The formula \( (uv)’ = u \cdot v’ + v \cdot u’ \) should be memorized for quick recall.

3. Work through multiple examples:

– Practice differentiating products of various functions to reinforce the concept.

4. Check your work:

– After applying the product rule, it’s helpful to re-evaluate the differentiation steps to avoid mistakes.

5. Understand the logic:

– Recognize that the product rule essentially distributes the differentiation operation across both functions, accounting for the change in each function.