Definition of the derivative Knowledge-Based Study

Study notes

1. From Secant Line to Tangent Line

   Consider a function $y = f(x)$. The average rate of change of $f$ over the interval $[x, x + h]$ is given by the slope of the secant line passing through the points $(x, f(x))$ and $(x + h, f(x + h))$:

   Average Rate of Change:

   $$\frac{\Delta y}{\Delta x} = \frac{f(x + h) - f(x)}{h}$$

   As the value of $h$ approaches 0, the second point slides along the curve toward the first point. The secant line approaches the tangent line to the curve at the point $x$. The slope of this tangent line represents the instantaneous rate of change of the function at that exact point.

   Formal Definition

   The derivative of a function $f$ with respect to $x$ is denoted by $f'(x)$ or $\frac{df}{dx}$. It is defined formally as the limit of the difference quotient as $h$ approaches 0, provided this limit exists:

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

   If this limit exists at a particular value of $x$, the function $f$ is said to be differentiable at $x$. If the limit does not exist, the derivative is undefined at that point.

   Derivative at a Point

   To evaluate the derivative of $f$ at a specific number $a$, we can compute $f'(a)$ using an alternative limit formulation:

   $$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

   This formulation expresses the slope of the secant line through $(a, f(a))$ and $(x, f(x))$ as $x$ approaches $a$. Both formulations are mathematically equivalent: substituting $x = a + h$ yields the original formula.

   Calculating a Derivative from the Definition

   Find the derivative of $f(x) = x^2 + 3x$ using the limit definition.

   Step 1: Set up the difference quotient

   $$\frac{f(x + h) - f(x)}{h} = \frac{((x + h)^2 + 3(x + h)) - (x^2 + 3x)}{h}$$

   Step 2: Expand and simplify the numerator

   $$(x^2 + 2xh + h^2 + 3x + 3h) - (x^2 + 3x) = 2xh + h^2 + 3h$$

   Step 3: Factor out h and cancel

   $$\frac{h(2x + h + 3)}{h} = 2x + h + 3 \quad \text{for } h \neq 0$$

   Step 4: Evaluate the limit as $h \to 0$

   $$f'(x) = \lim_{h \to 0} (2x + h + 3) = 2x + 3$$

   Therefore, the derivative of $f(x) = x^2 + 3x$ is $f'(x) = 2x + 3$.