Differentiation

Scheme of work: Year 12 A-Level: Pure 1: Differentiation

Prerequisite Knowledge

  • From GCSE
    • Find gradients of straight lines.
    • Expand and factorise algebraic expressions
    • Use the laws of indices to simplify expressions
    • Find the equation of a straight line.
    • Sketch quadratic graphs
    • Solve quadratic equations
  • From A-Level
    • Sketch cubic graphs
    • Solve cubic equations
    • Solve quadratic and linear inequalities

Success Criteria

  • Find the gradient of curves from first principals and understand this as differentiation.
  • Understand and use the derivative of f(x) as the gradient of the tangent to the graph of f(x) at a general point (xy).
  • Understand and use the derivative of f(x) as a rate of change.
  • Sketch the gradient function for a given curve.
  • Use differentiation to decide whether a function is increasing or decreasing.
  • Understand and use the second derivative as the rate of change of gradient 
  • Differentiate, for rational values of n, and related constant multiples, sums and differences
  • Apply differentiation to find gradients, tangents and normalmaxima and minima and stationary points 

Key Concepts

  • The process of finding of the gradient of the chord is called differentiation from first principles.  Where
\frac{dy}{dx} = \lim_{h \rightarrow 0} \left\{ \frac{f (x + h)-f(x)}{h}\right\}
  • The gradient or derivative of a curve at a point is the gradient of the tangent to the curve at that point.
  • When the graph increases from left to right, the gradient is positive. 
  • When the graph is decreasing from left to right, the gradient is negative.
  • If dy/dx is positive, the function is increasing, and if dy/dx is negative, then the function is decreasing.
  • When the tangent is horizontal, the gradient is zero, which represents a stationary point.
  • A stationary point on the curve f(x) is when f ‘ (x) = 0.  If f(x)  has a stationary point when x = a, then:
    • a minimum occurs if f'(a) > 0
    • a maximum occurs if f’ (a) < 0
  • Rules for differentiation
y = c\Longrightarrow
\frac{dy}{dx} =0
y = kx\Longrightarrow
\frac{dy}{dx} =k
y = kx^n\Longrightarrow
\frac{dy}{dx} =nkx^{n-1}
y = f(x) + g(x) \Longrightarrow\frac{dy}{dx} =f'(x)+g'(x)
  • Differentiating a function y = f(x) twice gives you the second-order derivative, f’ (x).  This tells you the rate of change of the gradient and is useful for identifying maximum and minimum points.

Common Misconceptions

  • A small percentage of students confuse differentiation with integration when answering exam questions.
  • Some students use the second derivative to determine whether a function is increasing or decreasing when they should use f(x) > 0 or f(x) < 0.
  • When asked to find the gradient of a tangent to a point on a curve, some students incorrectly make the gradient of the curve equal to zero and attempt to find x.
  • Students can struggle knowing the conditions for maxima and minima turning points.
  • Some students lose marks in their differentiation by not dropping the constant in the original function and not simplifying surds.
  • When asked to find maximum and minimum turning points, some students substituted a value of x on either side of f'(x)=0, which requires more work than using the second derivatives.
  • Exam questions often link applying differentiation to volume and surface area. As a result, some students lose marks deriving the volume or surface area equations, leading to incorrect derivatives.

Differentiation Resources

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