# 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.
• 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

#### Sequences and Series

Edexcel A-Level Mathematics Year 2: Pure 2: Algebraic Methods

#### T- Formulae

Scheme of work: A-Level Further Mathematics: Further Pure 1: The t – formulae

#### Regression, Correlation and Hypothesis Testing

A-Level Scheme of work: Edexcel A-Level Mathematics Year 2: Statistics: Regression, Correlation and Hypothesis Testing