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.
