Edexcel IGCSE Maths 复习笔记 3.11.1 Differentiation - Basics

Edexcel IGCSE Maths 复习笔记 3.11.1 Differentiation - Basics

What is differentiation?

  • Differentiation is part of the branch of mathematics called Calculus
  • It is concerned with the rate at which changes takes place – so has lots of real‑world uses:
  • The rate at which a car is moving – ie. its speed
  • The rate at which a virus spreads amongst a population

 

3.11.1-Diff-Basics-Notes-fig1

 

  • To begin to understand differentiation you’ll need to understand gradient

 

 

Gradient

  • Gradient generally means steepness.
    • For example, the gradient of a road up the side of a hill is important to lorry drivers

     

 

3.11.1-Diff-Basics-Notes-fig2

  • On a graph the gradient refers to how steep a line or a curve is
    • It is really a way of measuring how fast y changes as x changes
    • This may be referred to as the rate at which changes

     

  • So gradient is a way of describing the rate at which change happens

 

 

 

Straight lines and curves

  • For a straight line the gradient is always the same (constant)
    • Recall y= mx + c, where m is the gradient (see Straight Lines - Finding Equations)

     

 

3.11.1-Diff-Basics-Notes-fig3 

 

  • For a curve the gradient changes as the value of x changes
  • At any point on the curve, the gradient of the curve is equal to the gradient of the tangent at that point
    • tangent is a straight line that touches the curve at one point

     

 

3.11.1-Diff-Basics-Notes-fig4 

  • The gradient function is an expression that allows the gradient to be calculated anywhere along a curve
  • The gradient function is also called the derivative

 

 

How do I find the gradient function or derivative?

  • This is really where the fun with differentiation begins!
  • The derivative (dy/dx) is found by differentiating y

 

3.11.1-Diff-Basics-Notes-fig5

 

  • This looks worse than it is!
  • For powers of x ...
  • STEP 1   Multiply by the power
  • STEP 2   Take one off the power

 

 

3.11.1-Diff-Basics-Notes-fig6a

  • This method applies to positive and negative integers
  • Negative powers arise with fractions and reciprocals

 

3.11.1-Diff-Basics-Notes-fig6b

 

 

How do I find the value of a gradient?

  • Substitute the x value into the expression for the derivative, and evaluate it

 

3.11.1-Diff-Basics-Notes-fig7 

Exam Tip

When differentiating long, awkward expressions write each step out fully and simplify afterwards.Take extra care when differentiating negative powers of x

Worked Example

3.11.1-Diff-Basics-Example-fig1

转载自savemyexam

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