Building blocks

## Problem

Here are the graphs of $y=\dfrac{1}{x}$ and $y=\dfrac{1}{x+1}$ with tangents drawn at points A and B.

What happens when you move the point A? (Be careful when the $x$-coordinate of A gets close to $-1$.)

1. How could you work out the gradient of $y=\dfrac{1}{x+1}$ at the point $(2,\tfrac{1}{3})$?

2. How can you differentiate $\dfrac{1}{x+1}$?

3. How could you adapt these approaches to work out the derivative of $\dfrac{1}{2x+1}$?

Can you suggest other examples where you could use a similar approach to find derivatives?

In this problem we have only composed the function $\dfrac{1}{x}$ with functions of the form $ax+b$. Could you apply any of these ideas to polynomial, rational or trigonometric functions?