A curve has equation y=(2-x)(1+x)+3, A line passes through the point (2,3) and the curve at a point with x coordinate 2+h. Find the gradient of the line. Then use that answer to find the gradient of the curve at (2,3), stating the value of the gradient

The gradient of the line is given by change in Y divided by change in X. To find the Y value of the point we must put 2+h into the y=(2-x)(1+x)+3. Doing this we get y=(2-(2+h))(1+(2+h))+3=(-h)(3+h)+3= -3h -h² +3.

So the change in Y is from 3 to (-3h -h² +3), the change between these two values is 3h+h²) (found by subtracting the second from the first, remember the order must be the same for Y and X)

The change in x is simply the second subtracted from the first (like with the Y values), so the change in x = 2 -(2+h) = -h

So the gradient is (3h+h²)/-h = -h-3

We can use this to find the gradient at (2,3), if we imagine h getting smaller and smaller (tending to 0) then the line of (2,3) to the point with x coordinate 2+h becomes a line from (2,3) to a point incredibly close to (2,3), so close it tends to it and we can say it is (2,3), not just its close anymore.

So as h tends to 0, the gradient of the line in the first part becomes the gradient of the curve at (2,3), so the gradient is -h-3, as h tends to 0 the gradient becomes -3