Derivatives Fundamentals

Derivatives, a core concept in calculus, measure how a function changes, like the speed of a moving car given its position over time. Mathematically, a derivative is the slope of a function’s tangent line at a point, crucial for fields like physics, economics, and engineering. This MathMultiverse guide simplifies derivatives with examples, the power rule, interactive graphs, and practical applications, making the concept accessible and engaging.

Definition

The derivative of a function \( f(x) \) at point \( x \) represents its instantaneous rate of change, defined as:

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

This limit describes the slope of the tangent line at \( x \). As \( h \) approaches 0, the secant line becomes the tangent, revealing the function’s sensitivity to changes in \( x \).

Examples

Let’s compute derivatives for \( f(x) = x^2 \), \( x^3 \), \( \sqrt{x} \), \( \frac{1}{x} \), and \( x^5 \) using the limit definition.

\( f(x) = x^2 \)

Using the limit definition:

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

Derivative: \( f'(x) = 2x \). Slopes:

  • At \( x = 1 \): \( 2 \times 1 = 2 \).
  • At \( x = 2 \): \( 2 \times 2 = 4 \).
  • At \( x = -1 \): \( 2 \times (-1) = -2 \).
The linear slope reflects the parabola’s increasing steepness.

\( f(x) = x^3 \)

Calculating:

\[ f'(x) = \lim_{h \to 0} \frac{(x+h)^3 - x^3}{h} = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} = \lim_{h \to 0} (3x^2 + 3xh + h^2) = 3x^2 \]

Derivative: \( f'(x) = 3x^2 \). Slopes:

  • At \( x = 1 \): \( 3 \times 1^2 = 3 \).
  • At \( x = 2 \): \( 3 \times 2^2 = 12 \).
  • At \( x = 0 \): \( 0 \).
The quadratic growth mirrors the cubic’s rapid steepening.

\( f(x) = \sqrt{x} \)

For \( f(x) = x^{1/2} \):

\[ f'(x) = \lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}} = \lim_{h \to 0} \frac{x+h - x}{h (\sqrt{x+h} + \sqrt{x})} = \lim_{h \to 0} \frac{1}{\sqrt{x+h} + \sqrt{x}} = \frac{1}{2\sqrt{x}} \]

Derivative: \( f'(x) = \frac{1}{2\sqrt{x}} \). Slopes:

  • At \( x = 4 \): \( \frac{1}{2 \times 2} = \frac{1}{4} \).
  • At \( x = 1 \): \( \frac{1}{2 \times 1} = \frac{1}{2} \).
  • At \( x = 9 \): \( \frac{1}{2 \times 3} = \frac{1}{6} \).
The decreasing slope reflects the curve’s flattening.

\( f(x) = \frac{1}{x} \)

For \( f(x) = x^{-1} \):

\[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} = \lim_{h \to 0} \frac{\frac{x - (x+h)}{(x+h)x}}{h} = \lim_{h \to 0} \frac{-1}{(x+h)x} = -\frac{1}{x^2} \]

Derivative: \( f'(x) = -\frac{1}{x^2} \). Slopes:

  • At \( x = 1 \): \( -\frac{1}{1^2} = -1 \).
  • At \( x = 2 \): \( -\frac{1}{2^2} = -\frac{1}{4} \).
  • At \( x = -1 \): \( -1 \).
The negative slope indicates a decreasing hyperbola.

\( f(x) = x^5 \)

For a higher-degree polynomial:

\[ f'(x) = \lim_{h \to 0} \frac{(x+h)^5 - x^5}{h} = \lim_{h \to 0} \frac{5x^4h + 10x^3h^2 + 10x^2h^3 + 5xh^4 + h^5}{h} = \lim_{h \to 0} (5x^4 + 10x^3h + \ldots) = 5x^4 \]

Derivative: \( f'(x) = 5x^4 \). Slopes:

  • At \( x = 1 \): \( 5 \times 1^4 = 5 \).
  • At \( x = 2 \): \( 5 \times 2^4 = 80 \).
  • At \( x = 0 \): \( 0 \).
The rapid slope increase reflects the steepness of \( x^5 \).

Power Rule

The power rule simplifies differentiation of power functions:

\[ f(x) = x^n \implies f'(x) = n x^{n-1} \]

Verification with examples:

  • \( x^2 \): \( n = 2 \), \( f'(x) = 2 x^{2-1} = 2x \).
  • \( x^3 \): \( n = 3 \), \( f'(x) = 3 x^{3-1} = 3x^2 \).
  • \( \sqrt{x} = x^{1/2} \): \( n = \frac{1}{2} \), \( f'(x) = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}} \).
  • \( \frac{1}{x} = x^{-1} \): \( n = -1 \), \( f'(x) = -1 x^{-2} = -\frac{1}{x^2} \).
  • \( x^5 \): \( n = 5 \), \( f'(x) = 5 x^{5-1} = 5x^4 \).
  • \( x^{-3} \): \( n = -3 \), \( f'(x) = -3 x^{-4} = -\frac{3}{x^4} \).

This rule streamlines calculations across various exponents.

Graphical View

Visualizing derivatives as tangent slopes clarifies their behavior.

\( y = x^2 \)

Derivative: \( f'(x) = 2x \). At \( x = 1 \), slope = 2.

Parabola with tangent at \( x = 1 \).

\( y = x^3 \)

Derivative: \( f'(x) = 3x^2 \). At \( x = 1 \), slope = 3.

Cubic with tangent at \( x = 1 \).

\( y = \sqrt{x} \)

Derivative: \( f'(x) = \frac{1}{2\sqrt{x}} \). At \( x = 1 \), slope = \( \frac{1}{2} \).

Square root with tangent at \( x = 1 \).

Applications

Derivatives drive insights across disciplines.

  • Physics - Velocity and Acceleration: Position \( s(t) = 4t^2 + 2t \). Velocity: \( v(t) = s'(t) = 8t + 2 \). At \( t = 3 \), \( v = 26 \) units/s. Acceleration: \( a(t) = v'(t) = 8 \).
  • Optimization - Maximum Volume: Box volume \( V = x^2 h \), surface area \( x^2 + 4xh = 24 \). Solve \( h = \frac{24-x^2}{4x} \), so \( V = 6x - \frac{x^3}{4} \). Derivative: \( V'(x) = 6 - \frac{3x^2}{4} \). Set \( V'(x) = 0 \): \( x^2 = 8 \), \( x = 2\sqrt{2} \).
  • Economics - Marginal Revenue: Revenue \( R(x) = 50x - 0.5x^2 \). Marginal revenue: \( R'(x) = 50 - x \). At \( x = 20 \), \( R'(x) = 30 \).
  • Biology - Population Growth: Population \( P(t) = 500e^{0.03t} \). Growth rate: \( P'(t) = 15e^{0.03t} \). At \( t = 0 \), rate = 15.
  • Engineering - Beam Deflection: Deflection \( y(x) = \frac{w}{24EI} x^4 \). Slope: \( y'(x) = \frac{w}{6EI} x^3 \).