Differentiation basics:
The basic formula (in formula booklet):
y = a x n then
dy
= an x n−1
dx
Remember:
- If you differentiate a constant it is equal to zero.
4
you MUST rewrite it as y = 4x −3.
3
x
dy
- Notation: d x can also be written as y′ or f′(x) if using function notation.
dy
- d x is called the derivative.
- To differentiate y =
To find the gradient of the curve at a certain point:
1. Differentiate the curve
2. Substitute the given x value
(This also finds the gradient of the tangent at that point.)
To find the values of x for which a curve has a certain gradient:
1. Differentiate the curve
2. Make the derivative equal to the given value
3. Solve the resulting equation
Optimisation problems:
When the derivative is equal to zero, the function or curve has a maximum
or a minimum.
When you solve optimisation problems and you are asked to find the
“optimal/maximal/minimal” of some mathematical model, you need to use
the fact that
dy
= 0.
dx