Derivatives Game (Product Rule)
To use the product rule, you find the derivative of a product of two functions by taking the derivative of the first function, multiplying it by the second, and adding the first function multiplied by the derivative of the second. The formula is: if \(h(x)=f(x)g(x)\), then \(h^{\prime }(x)=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)\). Scroll down the page for a more detailed explanation.
 
In this game, you will need to find the derivative of the product of a polynomial function and a trigonometric function. Select one of the four possible answers. The game will tell you if you’re correct. If you are wrong, the correct answer will be highlighted in green. It includes a scoring system, and clear feedback to help you master this skill.
 

How to Play the Derivatives Game (Product Rule)
The game will show a you a product of two binomial functions. Your task is to differentiate them and select the correct answer.
Here’s how to play:

1. Start: Each Quiz consists of 10 questions. Click “Start Game”.
   
2. Look at the Problem: You will be given a product of a polynomial and a trig function.
   
3. Select Your Answer: Select the correct derivative of the product.br>
4. Check Your Work: Click “Submit Selection”. The game will tell you if you’re correct. If you are wrong, the correct answer will be highlighted in green.
   
5. Get a New Problem: Click “Next Question” for a new problem.
   Your score is tracked, showing how many you’ve gotten right.
   
6. Finish Game When you have completed 10 questions, click “Finish Game” to get your final score.
    
   

The Product Rule: Derivative of a Product
The Product Rule is used to find the derivative of a function that is the product of two or more functions.
The Rule
If you have a function \(f(x)\) that is the product of two functions, \(u(x)\) and \(v(x)\):
\(f(x) = u(x) \cdot v(x)\)
The derivative of \(f(x)\), denoted \(f’(x)\) or \(\frac{d}{dx}[f(x)]\), is given by:
\(f’(x) = u’(x)v(x) + u(x)v’(x)\)
In words: “The derivative of the first function times the second function, plus the first function times the derivative of the second function.”
 
Step-by-Step Example
Let’s find the derivative of the function:
\(y = x^2 \sin(x)\)
Step 1: Identify the two functions (\(u\) and \(v\))
Separate the given function \(y\) into \(u(x)\) and \(v(x)\).
Let the first function be \(u(x)\):
\(u(x) = x^2\)
Let the second function be \(v(x)\):
\(v(x) = \sin(x)\)
Step 2: Find the derivative of each function (\(u’\) and \(v’\))
Find the derivative of \(u’\) and \(v’\) separately.
Find the derivative of \(u’\) using the Power Rule:
\(u’(x) = \frac{d}{dx}[x^2] = 2x\)
Find the derivative of \(v’\):
\(v’(x) = \frac{d}{dx}[\sin(x)] = \cos(x)\)
Step 3: Apply the Product Rule formula
Substitute \(u\), \(v\), \(u’\), and \(v’\) into the Product Rule formula:
\(f’(x) = u’(x)v(x) + u(x)v’(x)\)
Substitute the expressions:
\(\frac{dy}{dx} = (2x) \cdot (\sin(x)) + (x^2) \cdot (\cos(x))\)
Step 4: Simplify (if necessary)
The final derivative is:
\(\frac{dy}{dx} = 2x\sin(x) + x^2\cos(x)\)
 

This video gives a clear, step-by-step approach to learn how to find the derivative using the product rule.

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