Dynamic Calculus
Home > The gradient function
The derived function (Part 3)
The applet below shows the graph of the function f(x) = x³.
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
- Reposition the point \(P\) by moving the 'slider' left and right. What is the significance of the red point?
- If \(f(x) = {x^3}\), find \(f'(x)\). Check your answer by clicking the 'checkbox'.
- Where does the graph of \(y = f'(x)\) touch the \(x\)-axis? How is this related to the gradient of the curve \(y = f(x)\) at the point?
- What is the range of \(y = f'(x)\)? Explain the significance of this with regard to the gradient of \(y = f(x)\).
- The function \(f(x) = {x^3}\) is an 'odd' function. Why? What kind of function is \(y = f'(x)\)? Why?
- What is the degree of the function \(f(x) = {x^3}\)? What is the degree of the derived function \(y = f'(x)\)?
- What would be the effect on the graph of \(y = f'(x)\) of moving \(y = f(x)\) 'up the \(y\)-axis'? What would be the effect on the graph of \(y = f'(x)\) of moving \(y = f(x)\) 'down the \(y\)-axis'?
- Give an example of another function whose derivative is \(f'(x) = 3{x^2}\). In your book draw a sketch of the function and its derived function.