Differentiating Powers of x (DP IB Applications & Interpretation (AI)): Revision Note

Differentiating Powers of x

What is differentiation?

• Differentiation is the process of finding an expression of the derivative (gradient function) from the expression of a function

How do I differentiate powers of x?

• Powers of  are differentiated according to the following formula:

  • If then where 

  • This is given in the formula booklet

• If the power of  is multiplied by a constant then the derivative is also multiplied by that constant

  • If then where  and  is a constant

• The alternative notation (to) is to use 

  • If  then 

    • e.g.  If  then 

• Don't forget these two special cases:

  • If then

    • e.g.  If  then 

  • If then

    • e.g.  If  then 

  • These allow you to differentiate linear terms in  and constants

• Functions involving roots will need to be rewritten as fractional powers of  first

  • e.g.  If  then rewrite as and differentiate

• Functions involving fractions with denominators in terms of  will need to be rewritten as negative powers of  first

  • e.g.  If then rewrite as and differentiate

How do I differentiate sums and differences of powers of x?

•  The formulae for differentiating powers of  apply to all rational powers so it is possible to differentiate any expression that is a sum or difference of powers of 

  • e.g.  If then
    
    

• Products and quotients cannot be differentiated in this way so would need expanding/simplifying first

  • e.g.  If then expand to which is a sum/difference of powers of  and can be differentiated