Lecture: Derivatives as Limits — The Algebraic Definition

Objective

To introduce the formal definition of the derivative using limits, and apply it to compute derivatives of well-known functions. This builds on the intuition of average and instantaneous rate of change.

Motivation Review: From Average to Instantaneous

Recall from our last lecture:

• The average rate of change between two points on a function:

  $$\text{Average Rate} = \frac{f(x+h) - f(x)}{h}$$

• As $h \to 0$, we approach the instantaneous rate of change, or the derivative:

  $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

This is called the difference quotient. It's the heart of differential calculus.

Algebraic Definition of the Derivative

Let $f(x)$ be a function. The derivative of $f$ at a point $x$ is:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

If this limit exists, $f$ is said to be differentiable at $x$.

This limit represents the slope of the tangent line to the graph of $f$ at $x$.

Notation

There are several ways to denote the derivative of a function. All represent the same concept: the instantaneous rate of change or the slope of the tangent line.

1. Leibniz Notation

Emphasizes the variables involved:

$$\frac{dy}{dx}, \quad \frac{df}{dx}, \quad \frac{d}{dx}[f(x)]$$

This notation is especially useful in applied problems (e.g., physics) and when working with related rates or implicit differentiation.

2. Lagrange Notation

Commonly used in pure mathematics:

$$f'(x), \quad y'$$

This is compact and often used when the independent variable is clear from context.

3. Newton Notation

Used mainly in physics and engineering for derivatives with respect to time:

$$\dot{y}, \quad \ddot{y}$$

Where $\dot{y}$ represents the first derivative with respect to time (velocity, for example), and $\ddot{y}$ the second derivative (acceleration).

4. Prime Notation for Higher Derivatives

• First derivative: $f'(x)$
• Second derivative: $f''(x)$
• $n$-th derivative: $f^{(n)}(x)$

For example:

$$\frac{d^2y}{dx^2}, \frac{d^2f}{dx^2}, \frac{d^2}{dx^2}[f(x)]$$

Step-by-Step Example: $f(x) = x^2$

Let us compute $f'(x)$ using the limit definition:

$$f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h}$$

Expand:

$$= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h}$$

$$= \lim_{h \to 0} (2x + h) = 2x$$

So, $f'(x) = 2x$.

Another Example: $f(x) = \frac{1}{x}$

$$f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h}$$

Use common denominator:

$$= \lim_{h \to 0} \frac{\frac{x - (x + h)}{x(x + h)}}{h} = \lim_{h \to 0} \frac{-h}{hx(x + h)}$$

Cancel $h$:

$$= \lim_{h \to 0} \frac{-1}{x(x + h)} = -\frac{1}{x^2}$$

So, $f'(x) = -\frac{1}{x^2}$

Additional Examples

1. $f(x) = \sqrt{x}$

$$f'(x) = \lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h}$$

Multiply by the conjugate:

$$= \lim_{h \to 0} \frac{x + h - x}{h(\sqrt{x + h} + \sqrt{x})} = \lim_{h \to 0} \frac{1}{\sqrt{x + h} + \sqrt{x}} = \frac{1}{2\sqrt{x}}$$

2. $f(x) = |x|$

$$f'(x) = \lim_{h \to 0} \frac{|x + h| - |x|}{h}$$

at $x = 0$, we have:

$$\lim_{h \to 0} \frac{|0 + h| - |0|}{h} = \lim_{h \to 0} \frac{|h|}{h}$$

Looking at the left and right limits, we have:

$$\lim_{h \to 0^-} \frac{|h|}{h} = -1 \quad \text{and} \quad \lim_{h \to 0^+} \frac{|h|}{h} = 1$$

Since the left and right limits are not equal, the limit does not exist.

This limit does not exist at $x = 0$. So $f(x) = |x|$ is not differentiable at $x = 0$.

Famous Derivatives

Exercises

1. Find derivatives

Using the limit definition, find the derivative of the following functions:

• (a) $f(x) = x^3$

• (b) $f(x) = \sqrt{x}$

• (c) $f(x) = 2x + 5$

• (d) $f(x) = 3x^4$

• (e) $f(x) = x^{-2}$

• (f) $f(x) = 5$

2. Prove from the limit definition:

• (a) $f(x) = \frac{1}{x} \Rightarrow f'(x) = -\frac{1}{x^2}$
• (b) $f(x) = x^2 + 3x \Rightarrow f'(x) = 2x + 3$

3. Find $f'(x)$ using the limit definition.

• (a) $f(x) = \frac{1}{x^2 + 1}$
• (b) $f(x) = \frac{1}{\sqrt{x}}$

4. logarithmic function

Derive the formula for $f(x) = \ln(x)$ from the limit definition:

$$f'(x) = \lim_{h \to 0} \frac{\ln(x + h) - \ln(x)}{h}$$

Use the identity: $\ln(x + h) - \ln(x) = \ln\left(1 + \frac{h}{x}\right)$, and the definition of $e$.

5. absolute value function

Let $f(x) = |x - 1|$. Determine where $f$ is not differentiable and explain why using the limit definition.

Extra Advanced Exercises:

1. Piecewise Function Differentiability

Let:

$$f(x) = \begin{cases} x^2 + 2x, & x < 1 \\ ax + b, & x \geq 1 \end{cases}$$

Find values of $a$ and $b$ such that:

• (a) $f$ is continuous at $x = 1$
• (b) $f$ is differentiable at $x = 1$

2. Challenge with Radical and Rational Function

Use the limit definition to find $f'(x)$ for:

$$f(x) = \frac{x}{\sqrt{x + 1}}$$

3. Limit Form Involving Trigonometry

Use the identity:

$$\lim_{h \to 0} \frac{\sin(h)}{h} = 1$$

to derive:

$$\frac{d}{dx}(\sin(x)) = \cos(x)$$

4. Absolute and Rational Hybrid

Let:

$$f(x) = \frac{|x|}{x}$$

• (a) Use the limit definition to determine where $f(x)$ is differentiable.
• (b) Graph it and explain why the derivative does or does not exist at certain points.

5. Pointwise Derivative of a Nowhere-Differentiable Function

Read about the Weierstrass function:

$$f(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x) \quad \text{with } 0 < a < 1, \ b \text{ odd, } ab > 1 + \frac{3\pi}{2}$$

• (a) Explain why this function is continuous everywhere
• (b) Investigate why it is not differentiable anywhere
• (c) Attempt a limit-based justification (conceptual only)

Wrap-Up

• Derivatives measure instantaneous change, defined using limits
• The limit definition is foundational, even if we later use rules to shortcut the process
• Not all functions are differentiable everywhere (e.g., piecewise, absolute value, radicals at zero)