Derivative (Trig Function) Game
Memorizing these six basic derivative rules is essential for calculus. Scroll down the page for a more detailed explanation.
 

 
In this game, you will need to find the derivative of a function using the Trig Function and other derivative rules. Select one of the four possible answers. The correct answer will be highlighted in green. It includes a scoring system, and clear feedback to help you master this skill.
 

Trig Derivative Challenge

Progress Score: 0

Find the Derivative

Next Problem ➜

 

How to Play the Derivative (Trig Function) Game
The game will show you a function. Your task is to differentiate the trig function 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 function.
   
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.
    
   

The Trig Function
Memorizing these six basic derivative rules is essential for calculus. \(f(x)=\sin(x)\) \(f’(x)=\cos(x)\) Mnemonic Tip: S-C (Sine to Cosine)

\(f(x)=\cos(x)\) \(-\sin(x)\) Mnemonic Tip: Co-functions ($\cos, \cot, \csc$) always yield a negative derivative.

$\tan(x)$

$\sec^2(x)$

Quotient Rule

Tangent goes to Secant squared.

$\cot(x)$

$-\csc^2(x)$

Quotient Rule

Co-function $\rightarrow$ Negative. Cotangent goes to Cosecant squared.

$\sec(x)$

$\sec(x) \tan(x)$

Quotient Rule

Secant finds itself and tangent.

$\csc(x)$

$-\csc(x) \cot(x)$

Quotient Rule

Co-function $\rightarrow$ Negative. Cosecant finds itself and cotangent.
Step-by-Step Procedure
To find the derivative of \(f(x) = \frac{u(x)}{v(x)}\):
Step

1. Identify
   Identify \(u(x)\) (high) and \(v(x)\) (low).
   
2. Differentiate
   Find the derivatives \(u’(x)\) and \(v’(x)\).
   
3. Substitute
   Substitute all four components into the formula: \(\frac{v u’ - u v’}{v^2}\).
   
4. Simplify
   Simplify the numerator to get the final derivative.
    
   Detailed Example
   Problem: Find the derivative of \(f(x) = \frac{3x - 5}{2x^2 + 1}\).
   Step 1: Identify u and v
   \(u(x) = 3x - 5\) (High)
   \(v(x) = 2x^2 + 1\) (Low)
   Step 2: Find the Derivatives (u’ and v’)
   \(u’(x) = 3\) (dee-High)
   \(v’(x) = 4x\) (dee-Low)
   Step 3: Apply the Trig Function Formula
   \(f’(x) = \frac{\overbrace{(2x^2 + 1)}^{v} \overbrace{(3)}^{u’} - \overbrace{(3x - 5)}^{u} \overbrace{(4x)}^{v’}}{\underbrace{(2x^2 + 1)^2}_{v^2}}\)
   Step 4: Simplify the Result
   First, expand the numerator:
   \(\text{Numerator} = (6x^2 + 3) - (12x^2 - 20x)\)
   Distribute the subtraction sign:
   \(\text{Numerator} = 6x^2 + 3 - 12x^2 + 20x\)
   Combine like terms:
   \(\text{Numerator} = -6x^2 + 20x + 3\)
   Final Derivative:
   \(f’(x) = \frac{-6x^2 + 20x + 3}{(2x^2 + 1)^2}\)
    
   

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

• Show Video

 

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