Differentiating powers of x
The first type of function that we differentiate will be powers of x.
If attempting to prove these results (not done in this activity) we often begin by looking at positive integer powers and these are the first functions we may consider.
However, the rule works for negative and fractional powers of x as well.
Start by seeing if you can apply the rule to positive integer powers of x. Try to evaluate the derivative for different functions (change a and n with the sliders) and then check your answer.
[You can change the coordinates of the x-axis by changing x_min and x_max]
Differentiating positive integer powers.
Negative integer powers
Now use the following activity to notice that the same rule applies when working with negative integer powers.
Non-integer powers
Now try different (non-integer) values of n and see if you can correctly predict the derivative.