Differentiate these functions and compare the results with the original functions.

\[\begin{align} y&=(x+2)(x+3) \label{eq 1}\\ y&=(x+2)(x-1) \label{eq 2} \end{align}\]
  • What do you notice?

If you don’t notice anything, or you want to check out whether there is a pattern emerging you could try a few more examples of your own.

  • Can you create a rule to find the derivative of a function like this without expanding the brackets?

Since \(\eqref{eq 1}\) could be written \(y= x^2 +5x +6\) then \(\frac{dy}{dx} = 2x +5\)

and since \(\eqref{eq 2}\) could be written \(y=x^2 +x -2\) then \(\frac{dy}{dx} = 2x +1\).

If we compare these with the original functions we can see that each derivative is the sum of the two brackets that were multiplied in the original function.

  • Can you understand why this happens?
  • Will this always be the case?