Differentiation

This lesson covers Differentiation as required by the A-Level Mathematics Pure 1 specification. Differentiation is a fundamental tool of calculus that allows us to find the rate of change of a function. It has applications in finding gradients, tangents, normals, stationary points, and solving optimisation problems.

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The Gradient of a Curve

The gradient of a curve at any point is the gradient of the tangent to the curve at that point. Differentiation gives us a formula for this gradient.

The derivative of y = f(x) is written as:

dy/dx  or  f'(x)

It represents the rate of change of y with respect to x.

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Differentiating from First Principles

The derivative of f(x) from first principles is defined as:

f'(x) = lim (h → 0) [f(x + h) − f(x)] / h

Example: Differentiate f(x) = x² from first principles.

f(x + h) = (x + h)² = x² + 2xh + h²

f'(x) = lim (h → 0) [(x² + 2xh + h²) − x²] / h
      = lim (h → 0) [2xh + h²] / h
      = lim (h → 0) [2x + h]
      = 2x