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

*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

- 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 (
*x*,*y*). - 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,
*n,*and related constant multiples, sums and differences - Apply differentiation to find gradients, tangents and normal
*,*maxima and minima and stationary points

- 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.

- 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.

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