# Differentiation of a Product

$$9x^2 * 8x$$

## 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.