5.1.2 Definition of Derivatives

Definition of Derivatives

Differentiation is an operation that calculates the rate of change of a function with respect to a variable
- This means how much the function varies when the variable increases by one unit
To differentiate a function (f(x)) with respect to the variable x we use the notation
- $\frac{d}{d x} (f (x))$
The result is called the derivative

The rate of change of a function f(x) with respect to x can be thought of as the gradient function of the graph y = f(x)
- We can write the gradient function (or derivative) as
  - $f' (x)$ or $\frac{d y}{d x}$
The rate of change of a function (or the gradient of its graph) varies for different values of x
- For a linear function f(x) = mx + c the gradient is constant
  - We can write this as $f' (x) = m$ or $\frac{d y}{d x} = m$
- For the quadratic function f(x) = x² the gradient varies
  - Near the origin the gradient is close to 0
  - As x increases the gradient of the graph increases

The derivative of a function (or gradient of its graph) at a point is equal to the gradient of the tangent to the graph at that point
To estimate the gradient you could draw the tangent and calculate its gradient
To find the actual gradient at a point x
- Pick a second point on the curve close to the first point (call it x + h)
- Calculate the gradient of the chord joining the two points
  - $\frac{f (x + h) - f (x)}{(x + h) - x} = \frac{f (x + h) - f (x)}{h}$
- Move the second point closer to the first point (make h get close to zero)
- Examine what happens to the gradient of the chord
- The gradient of the tangent will be the limit of the gradients of the chords
  - $f^{'} (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$
  - You do not need to remember this formula