CIE A Level Maths: Pure 1復習筆記5.1.2 Definition of Derivatives
Definition of Derivatives
What is the derivative of a function??
- Differentiation?is an operation that calculates the?rate of change?of a?function?with respect to a variable
- This means how much the function varies when the variable increases by one unit
- To?differentiate?a function (f(x)) with respect to the variable?x?we use the?notation
The result is called the?derivative
What is the link between derivatives and gradients??
- The rate of change of a function f(x) with respect to?x?can be thought of as the?gradient?function?of the graph?y?= f(x)
- We can write the gradient function (or derivative) as
-
For the?quadratic function?f(x) =?x2 the gradient varies
- Near the origin the gradient is close to 0
- As?x?increases the gradient of the graph increases
How can I find the derivative of a function at a point??
- The?derivative?of a function (or gradient of its graph) at a point is equal to the?gradient of the tangent?to the graph at that point
- To?estimate?the gradient you could?draw the tangent?and calculate its?gradient
- To find the?actual gradient?at a point?x
- Pick a?second point?on the curve close to the first point (call it?x?+?h)
- Calculate the?gradient of the chord?joining the two points
You do not?need to remember this formula
Worked Example
Exam Tip
- Deriving a derivative from scratch is?not examinable
- This revision note is intended to give you an understanding of what derivatives do
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