Hyperbolic Functions
Hyperbolic functions (\(\sinh, \cosh, \tanh\), etc.) are defined using the natural exponential function, \(e^x\). They are analogous to trigonometric functions but are based on the hyperbola rather than the circle.
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 Hyperbolic functions using the Hyperbolic 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.
 

Hyperbolic Challenge

Progress Score: 0

Find the Derivative

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How to Play the Derivative (Hyperbolic) 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 Hyperbolic Functions
These are the rules you need to know:
\(f(x) = \sinh(x)\)
\(f’(x) = \cosh(x)\)

\(f(x) = \cosh(x)\)
\(f’(x) = \sinh(x)\)

\(f(x) = \tanh(x)\)
\(f’(x) = \text{sech}^2(x)\)

\(f(x) = \coth(x)\)
\(f’(x) = -\text{csch}^2(x)\)

\(f(x) = \text{sech}(x)\)
\(f’(x) = -\text{sech}(x) \tanh(x)\)

\(f(x) = \text{csch}(x)\)
\(f’(x) = -\text{csch}(x) \coth(x)\)

Applying the Chain Rule
Just like standard trig functions, if the argument is a function $u$ of $x$, you must multiply by the derivative of the inner function,\(\frac{du}{dx}\).

\(\frac{d}{dx}[\cosh(u)] = \sinh(u) \cdot \frac{du}{dx}\)

Example: Find the derivative of \(y = \tanh(x^3 + 5)\)
Identify \(u = x^3 + 5\)
Find \(\frac{du}{dx} = 3x^2\)
Apply the Rule:
\(y’ = \text{sech}^2(u) \cdot \frac{du}{dx}\)
\(y’ = \text{sech}^2(x^3 + 5) \cdot 3x^2\)
\(y’ = 3x^2 \text{sech}^2(x^3 + 5)\)
 

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

• Show Video

 

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