Differentiation from First Principles (OCR AS Maths A: Pure): Revision Note

Differentiation from first principles

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)

• 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

$f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}$

• $\underset{h\to 0}{\lim }$ means the 'limit as h tends to zero'

• When$h=0$, $\frac{f(x+h)-f\left(x\right)}{h}=\frac{f\left(x\right)-f\left(x\right)}{0}=\frac{0}{0}$ 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

  • e.g.  Show, from first principles, that the derivative of 3x² is 6x

  • $f\left(x\right)=3x^2$ so $f\text{'}\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{lim}\frac{3{(x+h)}^2-3x^2}{h^{}}$

• STEP 2
  Expand f(x+h) in the numerator

  • $f\text{'}\left(x\right)=\underset{h\to 0}{lim}\frac{3(x^2+2hx+h^2)-3x^2}{h}$

  • $f\text{'}\left(x\right)=\underset{h\to 0}{lim}\frac{3x^2+6hx+3h^2-3x^2}{h}$

• STEP 3
  Simplify the numerator, factorise and cancel h with the denominator

  • $f\text{'}\left(x\right)=\underset{h\to 0}{lim}\frac{h(6x+3h)}{h}$

• STEP 4
  Evaluate the remaining expression as h tends to zero

  • $f\text{'}\left(x\right)=\underset{h\to 0}{lim}(6x+3h)=6x$     $Ash\to 0,(6x+3h)\to (6x+0)\to 6x$

  • $\therefore$The derivative of $3x^2$ is $6x$