Introduction to Differentiation (DP IB Maths: AI HL)

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Introduction to Derivatives

  • Before introducing a derivative, an understanding of a limit is helpful

What is a limit?

  • The limit of a function is the value a function (of x) approaches as x approaches a particular value from either side
    • Limits are of interest when the function is undefined at a particular value
    • For example, the functionspace f left parenthesis x right parenthesis equals fraction numerator x to the power of 4 minus 1 over denominator x minus 1 end fraction will approach a limit as x approaches 1 from both below and above but is undefined at x equals 1 as this would involve dividing by zero

What might I be asked about limits?

  • You may be asked to predict or estimate limits from a table of function values or from the graph of y equals f left parenthesis x right parenthesis
  • You may be asked to use your GDC to plot the graph and use values from it to estimate a limit

What is a derivative?

  • Calculus is about rates of change
    • the way a car’s position on a road changes is its speed (velocity)
    • the way the car’s speed changes is its acceleration
  • The gradient (rate of change) of a (non-linear) function varies with x
  • The derivative of a function is a function that relates the gradient to the value of x
  • The derivative is also called the gradient function

How are limits and derivatives linked?

  • Consider the point P on the graph of y equals f left parenthesis x right parenthesis as shown below
    • left square bracket P Q subscript i right square bracket is a series of chords

5-1-2-definiton-of-derivatives-diagram-1

  • The gradient of the functionspace f left parenthesis x right parenthesis at the point P is equal to the gradient of the tangent at point P
  • The gradient of the tangent at point P is the limit of the gradient of the chords left square bracket P Q subscript i right square bracket as point Q ‘slides’ down the curve and gets ever closer to point P
  • The gradient of the function changes as x changes
  • The derivative is the function that calculates the gradient from the value x

What is the notation for derivatives?

  • For the function y equals f left parenthesis x right parenthesisthe derivative, with respect to x, would be written as

fraction numerator straight d y over denominator straight d x end fraction equals f apostrophe left parenthesis x right parenthesis

  • Different variables may be used
    • e.g. If V equals f left parenthesis s right parenthesis then  fraction numerator straight d V over denominator straight d s end fraction equals f apostrophe left parenthesis s right parenthesis

Worked example

The graph of y equals f left parenthesis x right parenthesis wherespace f left parenthesis x right parenthesis equals x cubed minus 2 passes through the points P left parenthesis 2 comma space 6 right parenthesis comma space A left parenthesis 2.3 comma space 10.167 right parenthesis comma space B left parenthesis 2.1 comma space 7.261 right parenthesis and C left parenthesis 2.05 comma space 6.615125 right parenthesis.

a)
Find the gradient of the chords left square bracket P A right square bracket comma space left square bracket P B right square bracket and left square bracket P C right square bracket.

5-1-1-ib-sl-ai-aa-we1-soltn-a

b)
Estimate the gradient of the tangent to the curve at the point P.

5-1-1-ib-sl-ai-aa-we1-soltn-b

 

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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 x are differentiated according to the following formula:
    • Ifspace f left parenthesis x right parenthesis equals x to the power of n thenspace f apostrophe left parenthesis x right parenthesis equals n x to the power of n minus 1 end exponent where n element of straight rational numbers
    • This is given in the formula booklet
  • If the power of x is multiplied by a constant then the derivative is also multiplied by that constant
    • Ifspace f left parenthesis x right parenthesis equals a x to the power of n thenspace f apostrophe left parenthesis x right parenthesis equals a n x to the power of n minus 1 end exponent where n element of straight rational numbers and a is a constant
  • The alternative notation (tospace f apostrophe left parenthesis x right parenthesis) is to use fraction numerator straight d y over denominator straight d x end fraction
    • If y equals a x to the power of n then fraction numerator straight d y over denominator straight d x end fraction equals a n x to the power of n minus 1 end exponent
      • e.g.  If y equals negative 4 x to the power of 1 half end exponent then fraction numerator straight d y over denominator straight d x end fraction equals negative 4 cross times 1 half cross times x to the power of 1 half minus 1 end exponent equals negative 2 x to the power of negative 1 half end exponent
  • Don't forget these two special cases:
    • Ifspace f left parenthesis x right parenthesis equals a x thenspace f apostrophe left parenthesis x right parenthesis equals a
      • e.g.  If y equals 6 x then fraction numerator straight d y over denominator straight d x end fraction equals 6
    • Ifspace f left parenthesis x right parenthesis equals a thenspace f apostrophe left parenthesis x right parenthesis equals 0
      • e.g.  If y equals 5 then fraction numerator straight d y over denominator straight d x end fraction equals 0
    • These allow you to differentiate linear terms in x and constants
  • Functions involving roots will need to be rewritten as fractional powers of x first
    • e.g.  If space f left parenthesis x right parenthesis equals 2 square root of x then rewrite asspace f left parenthesis x right parenthesis equals 2 x to the power of 1 half end exponent and differentiate
  • Functions involving fractions with denominators in terms of x will need to be rewritten as negative powers of x first
    • e.g.  Ifspace f left parenthesis x right parenthesis equals 4 over x then rewrite asspace f left parenthesis x right parenthesis equals 4 x to the power of negative 1 end exponent and differentiate

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

  •  The formulae for differentiating powers of x apply to all rational powers so it is possible to differentiate any expression that is a sum or difference of powers of x
    • e.g.  Ifspace f left parenthesis x right parenthesis equals 5 x to the power of 4 minus 3 x to the power of 2 over 3 end exponent plus 4 then
      space f apostrophe left parenthesis x right parenthesis equals 5 cross times 4 x to the power of 4 minus 1 end exponent minus 3 cross times 2 over 3 x to the power of 2 over 3 minus 1 end exponent plus 0
      space f apostrophe left parenthesis x right parenthesis equals 20 x cubed minus 2 x to the power of negative 1 third end exponent
  • Products and quotients cannot be differentiated in this way so would need expanding/simplifying first
    • e.g.  Ifspace f left parenthesis x right parenthesis equals left parenthesis 2 x minus 3 right parenthesis left parenthesis x squared minus 4 right parenthesis then expand tospace f left parenthesis x right parenthesis equals 2 x cubed minus 3 x squared minus 8 x plus 12 which is a sum/difference of powers of x and can be differentiated

Examiner Tip

  • A common mistake is not simplifying expressions before differentiating
    • The derivative of open parentheses x squared plus 3 close parentheses open parentheses x cubed minus 2 x plus 1 close parentheses can not be found by multiplying the derivatives of open parentheses x squared plus 3 close parentheses and open parentheses x cubed minus 2 x plus 1 close parentheses

Worked example

The functionspace f left parenthesis x right parenthesis is given by

space f left parenthesis x right parenthesis equals 2 x cubed plus fraction numerator 4 over denominator square root of x end fraction,  where x greater than 0

Find the derivative ofspace f left parenthesis x right parenthesis
5-1-1-ib-sl-ai-aa-extra-we2-soltn-a

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.