Advanced Differentiation Techniques and Applications

This section delves deeper into Differentiation Further Maths GCSE concepts, providing more complex examples and applications.

Highlight: Differentiation of a constant is always zero. d/dx(constant) = 0

Let's explore a comprehensive example that demonstrates various aspects of differentiation:

a) For f(x) = 4x^2 - 8x + 3, find the gradient when x = 3

First, we find the gradient function: f'(x) = 8x - 8

Then, we substitute x = 3: f'(3) = 8(3) - 8 = 24 - 8 = 16

Example: The gradient at x = 3 is 16

b) Find the coordinates of the point on the graph of y = f(x) where the gradient is 8

We set the gradient function equal to 8: 8x - 8 = 8 Solving this, we get x = 2

To find y, we substitute x = 2 into the original function: f(2) = 4(2)^2 - 8(2) + 3 = 16 - 16 + 3 = 3

Example: The coordinates are (2, 3)

c) Find the gradient of y = f(x) at the points where the curve meets the line y = 4x - 5

We set the original function equal to the line equation: 4x^2 - 8x + 3 = 4x - 5

Solving this quadratic equation, we get x = 1 or x = 2

Now, we can find the gradients at these points using the gradient function: At x = 1: f'(1) = 8(1) - 8 = 0 At x = 2: f'(2) = 8(2) - 8 = 8

These examples demonstrate how to apply differentiation in various scenarios, which is crucial for AQA Further Maths GCSE past papers and Further Maths GCSE practice papers.