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 for the derivative (gradient function) of a function from an expression for the 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  term 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 

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

  • e.g.  If then rewrite as before differentiating

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

•  For an expression that is a sum or difference of powers of , just differentiate term by term

  • e.g.  If then
    
    

• Products and quotients cannot be differentiated in this way so would need to be expanded/simplified first

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