Derivative (Exp, Log) Game
Memorizing these exponential and logarithmic 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 Exp and Log functions using the Exp and Log derivative rules 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.
 

Exp & Log Challenge

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Find the Derivative

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How to Play the Derivative (Exp, Log) Game
The game will show you a function. Your task is to differentiate the exponential or log 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.
    
   

1. Derivatives of Natural Exponential and Logarithmic Functions
These are the most fundamental rules you need to know:
\(f(x) = e^x\)
\(f’(x) = e^x\)
The derivative of \(e^x\) is itself.

\(f(x) = \ln(x)\)
\(f’(x) = \frac{1}{x}\)
The derivative of the natural logarithm of $x$ is the reciprocal of $x$.

2. Derivatives of General Exponential and Logarithmic Functions
These rules apply when the base is any positive constant \(a\) (where \(a \ne 1\)).
\(f(x) = a^x\)
\(f’(x) = a^x \cdot \ln(a)\)
It is the function itself multiplied by the natural logarithm of the base \(a\).

\(f(x) = \log_a(x)\)
\(f’(x) = \frac{1}{x \cdot \ln(a)}\)
It is \(\frac{1}{x}\) multiplied by the reciprocal of the natural logarithm of the base \(a\).

Note: If you substitute \(a=e\) into the general rules, you recover the natural rules, since \(\ln(e) = 1\):
\(\frac{d}{dx}[e^x] = e^x \cdot \ln(e) = e^x \cdot 1 = e^x\)
\(\frac{d}{dx}[\log_e(x)] = \frac{d}{dx}[\ln(x)] = \frac{1}{x \cdot \ln(e)} = \frac{1}{x \cdot 1} = \frac{1}{x}\)

3. Applying the Chain Rule (The Most Common Scenario)
In most problems, the derivative is taken with respect to a composite function, \(u\), not just \(x\). This requires the Chain Rule.
The Chain Rule states: \(\frac{d}{dx}[f(g(x))] = f’(g(x)) \cdot g’(x)\)

A. Natural Exponential Chain Rule
If \(y = e^{u}\), where \(u\) is a function of \(x\), then the derivative is:
\(\frac{d}{dx}[e^{u}] = e^{u} \cdot \frac{du}{dx}\)

Example: Find the derivative of \(y = e^{\tan(x)}\)

1. Identify \(u = \tan(x)\)
   
2. \(\frac{du}{dx} = \sec^2(x)\)
   
3. Apply the Rule:
   \(y’ = e^{\tan(x)} \cdot \sec^2(x)\)
   
   B. Natural Logarithm Chain Rule
   If \(y = \ln(u)\), where \(u\) is a function of \(x\), then the derivative is:
   \(\frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} = \frac{u’}{u}\)
   
   Example: Find the derivative of \(y = \ln(5x^3 - 2)\)
   Identify \(u = 5x^3 - 2\)
   \(\frac{du}{dx} = 15x^2\)
   Apply the Rule:
   \(y’ = \frac{1}{5x^3 - 2} \cdot (15x^2)\)
   \(y’ = \frac{15x^2}{5x^3 - 2}\)
    
   

This video gives a clear, step-by-step approach to learn how to find the derivative of a exponential or logarithmic function.

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