Dynamic Calculus

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Differentiation from first principles

The applet below investigates the formal definition of the derivative and differentiation from first principles.
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]

 

 

1. Look closely at the definition of the function \(g(x)\). How does this relate to what you have learned about the formal definition of the derivative?
2. Investigate the function of the 'slider' by moving it left and right. What do you observe about the position of \(y = g(x)\) as \(h\) approaches zero? What does this illustrate about the gradient function of \(y = f(x)\)?
3. Find the gradient function for \(f(x) = {x^2}\). Check your solution by clicking the 'checkbox'. What is the gradient of the curve at the point where \(x = 3\)?
4. Differentiate \(f(x) = {x^2} + 2x\) from first principles and then find the gradient of the curve at the point where \(x = 2\).

State of New South Wales, Department of Education and Training 2008, Created with GeoGebra