Derivatives

Concept Explanation

While algebra measures average rates of change over finite intervals ($m = \frac{\Delta y}{\Delta x}$), calculus measures instantaneous rates by taking the limit as the interval $\Delta x \to 0$. If this limit exists at $x$, the function is differentiable at that point.

Visual / Geometric Intuition

Interactive Desmos Graph

Engineering Applications

Derivatives are everywhere in engineering physics. Velocity is the derivative of position ($v = \frac{ds}{dt}$), acceleration is the derivative of velocity ($a = \frac{dv}{dt}$), and electric current is the derivative of charge flow ($I = \frac{dq}{dt}$).

Example Problem

Use the limit definition of the derivative to find $f'(x)$ for $f(x) = x^2$.

Solution

Apply the limit definition:

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

Expand the numerator:

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

Simplify the terms:

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

Factor out and cancel $h$:

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

Evaluate the limit as $h \to 0$:

$$f'(x) = 2x$$

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Connections & References

• Parent Concepts: Continuity
• Sub-concepts: Chain Rule, Implicit Differentiation, Partial Derivatives
• Course Links: Calc-1 Session 06