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What is the Proof of Product Rule?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The Proof of Product Rule is a fundamental concept in calculus that helps us find the derivative of a function that is formed by multiplying two other functions together. It states that if you have two functions, say u(x) and v(x), and you want to find the derivative of their product, (u(x) * v(x)), you don't just multiply their individual derivatives. Instead, you take the derivative of the first function multiplied by the second function, and then add the first function multiplied by the derivative of the second function.
Simple Example
Quick Example
Imagine you're selling two types of sweets, ladoos (function u) and jalebis (function v), and their popularity changes each day (x). If you want to know how your total sweet sales (product of ladoos and jalebis) are changing, you can't just multiply how fast ladoo sales are changing by how fast jalebi sales are changing. You need to consider how a change in ladoo sales affects the total, plus how a change in jalebi sales affects the total.
Worked Example
Step-by-Step
Let's find the derivative of the function f(x) = (x^2) * (sin(x)). Here, u(x) = x^2 and v(x) = sin(x).
Step 1: Find the derivative of u(x). If u(x) = x^2, then u'(x) = 2x.
---Step 2: Find the derivative of v(x). If v(x) = sin(x), then v'(x) = cos(x).
---Step 3: Apply the Product Rule formula: (u * v)' = u' * v + u * v'.
---Step 4: Substitute the functions and their derivatives into the formula.
---Step 5: (x^2 * sin(x))' = (2x) * (sin(x)) + (x^2) * (cos(x)).
---Step 6: Simplify the expression.
---Answer: The derivative of f(x) = x^2 * sin(x) is 2x sin(x) + x^2 cos(x).
Why It Matters
The Product Rule is crucial for understanding how complex systems change. Engineers use it to design efficient electric vehicles and analyze signals. In AI/ML, it helps train models by understanding how different factors contribute to a prediction, making it vital for careers in data science and machine learning.
Common Mistakes
MISTAKE: Students often assume the derivative of a product is simply the product of the derivatives, i.e., (uv)' = u'v'. | CORRECTION: Remember the Product Rule formula: (uv)' = u'v + uv'. You must add the two parts, not just multiply the derivatives.
MISTAKE: Forgetting to apply the chain rule when u or v are composite functions (functions within functions). | CORRECTION: Always find the derivative of u and v correctly, using the chain rule if needed, before plugging them into the Product Rule.
MISTAKE: Mixing up which function is differentiated and which is left as is in each term (e.g., doing u'v' + uv). | CORRECTION: The formula is u'v + uv'. One function is differentiated in the first term (u'), the other in the second term (v'), and the other function remains as is.
Practice Questions
Try It Yourself
QUESTION: Find the derivative of y = (3x) * (e^x). | ANSWER: 3e^x + 3xe^x
QUESTION: What is the derivative of f(x) = (x^3 + 2x) * (cos(x))? | ANSWER: (3x^2 + 2)cos(x) - (x^3 + 2x)sin(x)
QUESTION: If a company's profit P(t) is given by P(t) = (t^2 + 5) * (ln(t)), where t is time in months, find the rate of change of profit, P'(t). | ANSWER: 2t ln(t) + (t^2 + 5)/t
MCQ
Quick Quiz
Which of the following correctly represents the Product Rule for finding the derivative of h(x) = f(x) * g(x)?
h'(x) = f'(x) * g'(x)
h'(x) = f'(x) * g(x) + f(x) * g'(x)
h'(x) = f(x) * g(x) + f'(x) * g'(x)
h'(x) = f'(x) + g'(x)
The Correct Answer Is:
B
Option B correctly states the Product Rule: the derivative of the first function times the second, plus the first function times the derivative of the second. Options A, C, and D are incorrect applications or misinterpretations of the rule.
Real World Connection
In the Real World
In FinTech, banks use the Product Rule to analyze how the value of a financial product changes when multiple factors (like interest rates and market demand) are multiplied together. For example, if a loan's repayment amount depends on the principal and a changing interest rate, the Product Rule helps calculate how quickly the total payment changes with time. This helps in risk assessment and creating new investment strategies.
Key Vocabulary
Key Terms
DERIVATIVE: The rate at which a function changes at any given point | FUNCTION: A relationship where each input has exactly one output | PRODUCT: The result of multiplying two or more numbers or expressions | CALCULUS: A branch of mathematics dealing with rates of change and accumulation of quantities
What's Next
What to Learn Next
Next, you should explore the 'Quotient Rule,' which is used to find the derivative of a function that is a division of two other functions. It's another essential tool in calculus and builds directly on your understanding of how to differentiate complex expressions.
