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Differentiation from First Principles - Practice

Maths: Edexcel A Level Pure Maths, topic 7.1.3

What is Differentiation by First Principles?

Differentiation by first principles is a method used to find the general expression for the derivate of a function.

What is the derivative of a function?

What is the derivative of a function?

  • The derivative of f(x)f(x) is an expression that tells us exactly what the gradient of a given function is for any value of xx.
What is the equation for the derivative?

What is the equation for the derivative?

  • The derivative of a function is given by the equation:
  • f′(x)=lim⁡h→0f(x+h)−f(x)hf\prime(x)= \lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}
How do we find the gradient using the derivative?

How do we find the gradient using the derivative?

  • To find the gradient of a function at a point (x,f(x))(x,f(x)) we need to work out the values of f(x+h)f(x+h) and f(x)f(x).
  • We can then substitute this into the derivative equation.
  • Finally, we work out what the expression is equal to when hh tends to 0.
  • This gives the gradient at that point.
How do we find the limit as $$h\to0$$?

How do we find the limit as h→0h\to0?

  • As you take the limit as hh tends to zero, any constant terms that do not contain a factor of hh will stay the same:
  • For example, lim⁡h→05=5\lim\limits_{h\to0}5=5
  • Any terms that contain at least one factor of hh will tend to zero as hh tends to zero:
  • For example, lim⁡h→03h=lim⁡h→0h2=lim⁡h→02h8=0\lim\limits_{h\to0}3h=\lim\limits_{h\to0}h^2=\lim\limits_{h\to0}2h^8=0

Test yourself on Differentiation from First Principles - Practice

  • What is the limit of the following expression as hh tends to zero?
    5h2+2h+175h^2 + 2h+17

  • What is the limit of the following expression as hh tends to zero?
    30h2+15h30h^2 + 15h

  • What is the correct equation for the derivative of a function?

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