Differentiation from first principles - part 2
The Gradient Function
In the previous activity we looked at how to find the gradient of a function at a point. Now we will try and generalise this for a function.
We will look again at the quadratic function . We have already seen that we can pick an x-value (for example x=1) and find the tangent at this point but we may want to find the tangent in general.
When we do this instead of getting a single value for gradient we will find the gradient function. A function into which we can substitute any x-value and find the gradient of the corresponding tangent.
In the activity below, you can experiment with changing h and observe the dotted green line (an approximation to the tangent) at different points. See if you can find the gradient function for the quadratic .
Question 1
Try some different quadratic functions. Can you spot a pattern in the gradient function? Describe it below.
Question 2
Now try some cubic functions, again see if you can identify a pattern?
Question 3
Can you summarise how to find the gradient function for different powers of x?
Question 4
Experiment with different functions and write any observations below.