CIE A Level Maths: Pure 1复习笔记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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