General Definition of Derivative

Sahabat Smaridasa - Mathematical Concepts

In this article, we will discuss the General Definition of Derivative. To understand this concept, we first explore two key ideas: tangent lines and instantaneous velocity.

Tangent Line, Secant Line, and Normal Line

Gambar 1: Garis normal, garis secan (tali busur), dan garis singgung pada kurva

Gambar 2: Garis secan melalui A(a, f(a)) dan B(a+Δx, f(a+Δx)) serta garis singgung

Gambar 3: Ilustrasi Δx (perubahan x) dan Δy (perubahan y) pada kurva

Slope of Secant and Tangent Lines

The slope of the secant line through points \(A(a, f(a))\) and \(B(a + \Delta x, f(a + \Delta x))\) is:

\[ m_{AB} = \frac{f(a + \Delta x) - f(a)}{\Delta x} \]

As \(\Delta x \to 0\), the secant line approaches the tangent line. Thus, the slope of the tangent line at \(x = a\) is:

\[ m = \lim_{\Delta x \to 0} \frac{f(a + \Delta x) - f(a)}{\Delta x} \]

Instantaneous Velocity

If \(f(t)\) represents the position at time \(t\), then the instantaneous velocity at \(t = a\) is:

\[ v = \lim_{\Delta t \to 0} \frac{f(a + \Delta t) - f(a)}{\Delta t} \]

Definition of Derivative

The derivative of a function \(f\) at \(x = a\), denoted \(f'(a)\), is defined as:

\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]

provided the limit exists. Equivalently,

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

General derivative function: \(f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\)

Notation

• Newton: \(f'(x)\) or \(y'\) for first derivative; \(f''(x)\) or \(y''\) for second derivative.
• Leibniz: \(\frac{dy}{dx}\) or \(\frac{df}{dx}\) for first derivative; \(\frac{d^2y}{dx^2}\) for second derivative.

📘 Example 1 : Slope of Tangent Line

Find the slope of the tangent line to the curve \(f(x) = x^2\) at the point \((2, 4)\).

Solution:

\[ \begin{align*} f'(2) &= \lim_{h \to 0} \frac{f(2+h) - f(2)}{h} \\ &= \lim_{h \to 0} \frac{(2+h)^2 - 4}{h} \\ &= \lim_{h \to 0} \frac{4 + 4h + h^2 - 4}{h} \\ &= \lim_{h \to 0} \frac{4h + h^2}{h} = \lim_{h \to 0} (4 + h) = 4 \end{align*} \]

Thus, the slope of the tangent line is \(4\).

📘 Example 2 : Instantaneous Velocity

A particle moves along a straight line with position function \(f(x) = 6x^3 + x^2\) (in meters) after \(x\) seconds.

a) Find the average velocity over the interval \(2 \le x \le 3\).

b) Find the instantaneous velocity at \(x = 2\) seconds.

Solution:

a) Average velocity = \(\frac{f(3)-f(2)}{3-2} = \frac{(6\cdot27+9) - (6\cdot8+4)}{1} = (162+9) - (48+4) = 171 - 52 = 119\) m/s.

b) Instantaneous velocity at \(x=2\):

\[ \begin{align*} v &= \lim_{h \to 0} \frac{f(2+h) - f(2)}{h} \\ &= \lim_{h \to 0} \frac{[6(2+h)^3 + (2+h)^2] - [48+4]}{h} \\ &= \lim_{h \to 0} \frac{76h + 37h^2 + 6h^3}{h} = \lim_{h \to 0} (76 + 37h + 6h^2) = 76 \end{align*} \]

So the instantaneous velocity is \(76\) m/s.

📘 Example 3 : Derivative at a Point

Find \(f'(-2)\) for \(f(x) = x^2 - 3x\).

Solution:

\[ \begin{align*} f'(-2) &= \lim_{h \to 0} \frac{f(-2+h) - f(-2)}{h} \\ &= \lim_{h \to 0} \frac{[(-2+h)^2 - 3(-2+h)] - [4 + 6]}{h} \\ &= \lim_{h \to 0} \frac{(4 - 4h + h^2 + 6 - 3h) - 10}{h} \\ &= \lim_{h \to 0} \frac{h^2 - 7h}{h} = \lim_{h \to 0} (h - 7) = -7 \end{align*} \]

📘 Example 4 : General Derivative from Definition

Find \(f'(x)\) for the following functions using the limit definition:

a) \(f(x) = 5x - 2\)    b) \(f(x) = x^2 + 2x\)    c) \(f(x) = \sin x\)

Solution:

a) \(f'(x) = \lim_{h \to 0} \frac{5(x+h)-2 - (5x-2)}{h} = \lim_{h \to 0} \frac{5h}{h} = 5\).

b) \(f'(x) = \lim_{h \to 0} \frac{[(x+h)^2+2(x+h)] - (x^2+2x)}{h} = \lim_{h \to 0} \frac{2xh + h^2 + 2h}{h} = 2x + 2\).

c) \(f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h} = \lim_{h \to 0} \frac{2\cos(x+\frac{h}{2})\sin\frac{h}{2}}{h} = \cos x\).

📘 Example 5 : Non-differentiability

Show that \(f(x) = |x|\) is not differentiable at \(x = 0\).

Solution:

Check the left-hand and right-hand limits of the difference quotient:

\[ f'(0) = \lim_{x \to 0} \frac{|x| - 0}{x} = \lim_{x \to 0} \frac{|x|}{x} \]

For \(x > 0\), \(\frac{|x|}{x} = 1\). For \(x < 0\), \(\frac{|x|}{x} = -1\). Since the left and right limits are not equal, the limit does not exist. Thus, \(f(x) = |x|\) is not differentiable at \(x = 0\).

Relationship Between Differentiability and Continuity

If a function \(f\) is differentiable at \(x = a\), then \(f\) is continuous at \(x = a\). The converse is not necessarily true (e.g., \(f(x)=|x|\) is continuous but not differentiable at \(x=0\)).

Note: While the limit definition is fundamental, in practice we will use derivative rules for algebraic, trigonometric, and other functions to compute derivatives more efficiently.