Derivatives Game (Binomials)
To find the derivative of the product of two binomial functions, either use the Product Rule or multiply the two binomials first to get a single polynomial and use the Power Rule. Scroll down the page for a more detailed explanation.
 
In this game, you will need to find the derivative of the product of two binomial functions. 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 (Binomials)
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 polynomial.
   
3. Select Your Answer: Select the correct derivative.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.
    
   

How to find the derivative of the product of two binomials
The Product Rule: Derivative of a Product
The Product Rule is used when you need to find the derivative of a function that is the product of two separate functions, u(x) and v(x).
The Formula
If you have a function f(x) defined as:
\(f(x) = u(x)\cdot v(x)\)
The derivative of \(f(x)\), denoted as \(f’(x)\), is given by the formula:
\(\frac{d}{dx} [u(x)v(x)] = u’(x)v(x) + u(x)v’(x)\)
Or simply:
\(f’(x) = u’v + uv’\)
Where:
\(u’\) is the derivative of the first function, \(u\).
\(v’\) is the derivative of the second function, \(v\).
 
Step-by-Step Example
Let’s find the derivative of the product of two binomials.
Problem: Find the derivative of \(f(x) = (x^2 + 3x) (4x - 5)\).
Step 1: Identify u and v
Define the two binomials as u(x) and v(x):
\(u(x) = x^2 + 3x\)
\(v(x) = 4x - 5\)
Step 2: Find the Derivatives, u’ and v’
Use the Power Rule (\(\frac{d}{dx} [x^n] = nx^{n-1}\)) to find the derivative of each part.
Derivative of u (u’):
\(u’(x) = \frac{d}{dx}(x^2 + 3x) = 2x + 3\)
Derivative of v (v’):
\(v’(x) = \frac{d}{dx}(4x - 5) = 4\)
Step 3: Apply the Product Rule Formula
Substitute u, v, u’, and v’ into the Product Rule formula:
\(f’(x) = u’v + uv’\)
\(f’(x) = (2x + 3) (4x - 5) + (x^2 + 3x) (4)\)
Step 4: Simplify and Expand
Now, distribute the terms to simplify the expression:
First term (u’v):
\((2x + 3)(4x - 5) = 8x^2 - 10x + 12x - 15 = 8x^2 + 2x - 15\)
Second term (uv’):
\((x^2 + 3x)(4) = 4x^2 + 12x\)
Combine the terms:
\(f’(x) = (8x^2 + 2x - 15) + (4x^2 + 12x)\)
Final Answer:
\(f’(x) = 12x^2 + 14x - 15\)
 

Alternative Method: Expansion First
For two simple binomials like the example above, you could also multiply them out first and then take the derivative of the resulting polynomial.
1. Expand f(x):
\(f(x) = (x^2 + 3x)(4x - 5)\)
\(f(x) = 4x^3 - 5x^2 + 12x^2 - 15x\)
\(f(x) = 4x^3 + 7x^2 - 15x\)
2. Take the derivative:
\(f’(x) = \frac{d}{dx}(4x^3 + 7x^2 - 15x)\)
\(f’(x) = 12x^2 + 14x - 15\)
 

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

• Show Video

 

Derivative Challenge (Polynomials)
Find the derivatives of polynomials using the Power, Sum, Difference, and Constant Multiple rules.

Derivative Challenge (Binomials)
Find the derivatives of the product two binomial functions. Either use the Product Rule or multiply the two binomials first to get a single polynomial and use the Power Rule.

Derivative Challenge (Product Rule)
Find the derivatives of the product a polynomial and a trig function. Use the Product Rule.

Derivative Challenge (Quotient Rule)
Find the derivatives of a function using the Quotient Rule.

Derivative Challenge (Chain Rule)
Find the derivatives of a function using the Chain Rule.

Derivative Rules
Find the derivatives of a function using the Power Rule, Constant Rule, Sum/Difference Rule, Product Rule, Quotient Rule, Chain Rule, and the rules for trigonometric and exponential functions.

Derivative Challenge (Trig Function)
Find the derivatives of Trigonometric Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

Derivative Challenge (Exp, Log)
Find the derivatives of Exponential and Logarithmic Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

Derivative Challenge (Hyperbolic)
Find the derivatives of Hyperbolic Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

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