Edexcel A Level Maths: Pure:复习笔记7.1.2 First Principles Differentiation
First Principles Differentiation
What is the derivative or gradient function?
- For a curve?y?=?f(x)?there is an associated function called the?derivative?or?gradient function
- The derivative of f(x) is written as f'(x) or
- The derivative is a formula that can be used to find the gradient of y = f(x) at any point, by substituting the x coordinate of the point into the formula
- The process of finding the derivative of a function is called differentiation
- We differentiate a function to find its derivative
What is differentiation from first principles?
- Differentiation from?first principles?uses the?definition?of the?derivative?of a function?f(x)
- The definition is
means the 'limit as?h?tends to zero'- When
?which is?undefined
- Instead we consider what happens as?h?gets closer and closer to zero
- Differentiation?from first principles?means using that definition to show what the?derivative?of a function is
How do I differentiate from first principles?
STEP 1:?Identify the function?f(x)?and substitute this into the first principles formula
STEP 2:?Expand?f(x+h)?in the numerator
? 
STEP 3:?Simplify the numerator, factorise and cancel h with the denominator
? 
STEP 4:?Evaluate the remaining expression as?h?tends to zero
??? 
? ? ?
? ? ?Exam Tip
- Most of the time you will not use first principles to find the derivative of a function (there are much quicker ways!). However, you can be asked on the exam to demonstrate differentiation from first principles.
- Make sure you can use first principles differentiation to find the derivatives of?kx,?kx2?and?kx3?(where?k?is a constant).
Worked Example
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