A
Level Maths Notes: C2 – The Difference Quotient –
Differentiating From First Principles
The gradient of the
straight line between two
pints is given by
We
may generalize this for a curve to find the gradient at any point,
but we need to take into account that the curve is actually a curve.
In more basic maths we can draw a tangent at a point and find the
gradient of the tangent by using the above equation for any two
points on the tangent. In practice, when writing down the difference
quotient -shown below – we let the distance between the two points
tend to zero then the value of the difference quotient at the point
is the actual gradient of the tangent at that point and we can
write
or
The formula for the difference quotient for a function
at
a point
is
Example: Find the difference quotient for the function
