Lecture 20: Definition of the Derivative✅

Slope and Rate of Change

The derivative of a function f at a point x = a is the instantaneous rate of change of f at x = a.
Consider the graph of a function y = f(x).
Take a point a and its image f(a). This forms the point (a, f(a)).
Take an increment h for a, so consider a + h and its image f(a + h). This forms the point (a + h, f(a + h)).
The blue line joins the points (a, f(a)) and (a + h, f(a + h)).

The slope of the line joining the points (a, f(a)) and (a + h, f(a + h)) is given by:
\frac{\Delta y}{\Delta x} = \frac{f(a + h) - f(a)}{(a + h) - a} = \frac{f(a + h) - f(a)}{h}

What happens to the slope of the line as h approaches zero?
As h gets smaller, the point (a + h, f(a + h)) moves towards (a, f(a)).
The lines joining (a, f(a)) and (a + h, f(a + h)) start to resemble a tangent line.
The tangent line to a graph at a point is a line that touches the graph at exactly one point.
The limit as h approaches zero of the fraction \frac{f(a + h) - f(a)}{h}, if it exists, is the slope of the tangent line to the graph of f at the point (a, f(a)).
This is called the instantaneous rate of change of the function f at the point a.

Formally, the derivative of a function at a point x = a is defined as:
\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
If this limit exists, the function is said to be differentiable at the point x = a.

Example 1: Consider the function f(x) = x^2 and the point a = 1.
Calculate the derivative of f at the point 1:
\lim{h \to 0} \frac{f(1 + h) - f(1)}{h} = \lim{h \to 0} \frac{(1 + h)^2 - 1^2}{h}
\lim{h \to 0} \frac{1 + 2h + h^2 - 1}{h} = \lim{h \to 0} \frac{2h + h^2}{h} = \lim_{h \to 0} (2 + h) = 2
Example 2: Find the tangent line to y = x^2 at x = 1.
- The derivative is the slope of the tangent line.
- We have a point (1, f(1)) and the slope f'(1).
- The equation of the line is y - f(1) = f'(1)(x - 1).

Given the point (1, f(1)), where f(1) = 1^2 = 1, and the slope f'(1) = 2, the equation of the tangent line is:
y - 1 = 2(x - 1)
y = 2x - 2 + 1 = 2x - 1

The quotient \frac{f(a + h) - f(a)}{h} is called the difference quotient.
When the derivative is positive, the tangent line slants to the right of the vertical.
When the derivative is negative, the tangent line slants to the left of the vertical.
When the derivative is zero, the tangent line is horizontal.

The derivative of a function f usually takes different values at different points a.
The derivative is itself a function.
If we replace the number a in the definition by a variable x, we can define the derivative of f, denoted as f'(x), as a function of x:
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
This is over a subset of the domain of f for which the limit exists.

The derivative at a specific point is numerical, while the derivative as a function uses a variable.
The domain of the derivative function f' is all points x for which f'(x) exists and are part of the domain of f.

A function f is differentiable on an open interval (a, b) if the function is differentiable at every point x in that interval.

Find the derivative of f(x) = x^2 using the limit definition:
f'(x) = \lim{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim{h \to 0} \frac{(x + h)^2 - x^2}{h}
= \lim{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim{h \to 0} \frac{2xh + h^2}{h}
= \lim_{h \to 0} (2x + h) = 2x
Thus, the derivative of f(x) = x^2 is f'(x) = 2x.
We can check f'(1) = 2 as before.

f'(x) represents the rate of change of the function at each point where it's defined.
If f represents the position of an object moving along a straight path, then f' represents the velocity of the object.
If f(x) represents the displacement of a particle, then f' is the velocity.

If the derivative is positive on an interval, the function is increasing on that interval.
If the derivative is negative on an interval, the function is decreasing on that interval.
If the derivative is zero on an interval, the function is constant on that interval.

Consider a function f(x). On the interval (-\infty, a), f'(x) > 0, so f(x) increases.
On the interval (b, \infty), f'(x) < 0, so f(x) decreases.
On the interval (a, b), f'(x) = 0, so f(x) is constant.

If a function is not continuous (e.g., has a jump), then the function is not differentiable at that point.
If a function has a corner (e.g., absolute value of x + 1), then the function is not differentiable at that point.

Remember: a limit exists if and only if the left and right limits exist and are equal to that value l.
Consider f(x) = |x + 1|. Show that the derivative doesn't exist at x = -1 by showing that the left and right limits of the difference quotient are different:
\lim{x \to -1^-} \frac{f(x + h) - f(x)}{h} \neq \lim{x \to -1^+} \frac{f(x + h) - f(x)}{h}

Cusp: The derivative doesn't exist because you have a vertical tangent line (infinite slope).
Vertical Tangent Line: Infinite slope, so the limit doesn't exist.
A continuous function is not necessarily differentiable.
If a function is differentiable, then it means that the function is continuous.
If f is differentiable at a point a, then f is continuous at a point a.
If f is not continuous at a point, then f cannot be differentiable at that point.

When a function is differentiable, we call it smooth (continuous and has no corners or cusps).
Local linearity means that when we zoom in on the graph of a differentiable function, it starts to look like a line.