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

Mr Mathematics Blog

Problem-Solving with Angles in Polygons

How to teach problem solving with angles in polygons through scaffolding.

Planes of Symmetry in 3D Shapes

Planes of Symmetry in 3D Shapes for Key Stage 3/GCSE students.

Use isometric paper for hands-on learning and enhanced understanding.

GCSE Trigonometry Skills & SOH CAH TOA Techniques

Master GCSE Math: Get key SOH-CAH-TOA tips, solve triangles accurately, and tackle area tasks. Ideal for students targeting grades 4-5.