First Principles Differentiation (DP IB Analysis & Approaches (AA): HL): Revision Note

First principles differentiation

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 

• 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 

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

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

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

$\mathrm{As}h\to 0,(6x+3h)\to (6x+0)\to 6x$

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