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 $(a + h, f(a + h))$.

Slope of the Line

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

Limit as h Approaches Zero

• 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 $a$.

Definition of the Derivative

• 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 $x = a$.

Examples

• Example 1: Consider the function $f(x) = x^2$ and the point $a = 1$.
  Calculate the derivative of $f$ at the point 1:
  $\lim<em>{h \to 0} \frac{f(1 + h) - f(1)}{h} = \lim</em>{h \to 0} \frac{(1 + h)^2 - 1^2}{h}$
  $\lim<em>{h \to 0} \frac{1 + 2h + h^2 - 1}{h} = \lim</em>{h \to 0} \frac{2h + h^2}{h} = \lim_{h \to 0} (2 + h) = 2$
• 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 $y - f(1) = f'(1)(x - 1)$.

Equation of the Tangent Line

• 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:
  $y - 1 = 2(x - 1)$
  $y = 2x - 2 + 1 = 2x - 1$

Notes on the Derivative

• 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.

Derivative of a Function

• 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.

Distinction

• 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 $f$.

Differentiability on Intervals

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

Example

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

Notes

• $f'(x)$ 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.

Graphical Meaning of the Derivative Function

• 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.

Example

• 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.

Continuity and Differentiability

• 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.

Limits

• 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:
  $\lim<em>{x \to -1^-} \frac{f(x + h) - f(x)}{h} \neq \lim</em>{x \to -1^+} \frac{f(x + h) - f(x)}{h}$

Pathological Cases where Derivative Doesn't Exist

• 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.

Local Linearity

• 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.

Revision of Important Derivatives

• Derivative of a constant: 0
• Derivative of $x$: 1
• Derivative of $x^\alpha$: $\alpha x^{\alpha - 1}$ for $\alpha \in \mathbb{R}$
• Derivative of $e^x$: $e^x$
• Derivative of $\ln(x)$: $\frac{1}{x}$
• Derivative of $\sin(x)$: $\cos(x)$
• Derivative of $\cos(x)$: $-\sin(x)$
• Use the definition of $f'(x)$ to find these derivatives (first principles).