Calculus: Differentiation and Its Applications

What Is Differentiation?

Differentiation is a fundamental concept in calculus that deals with the rate of change of a function. The derivative represents the slope or instantaneous rate of change at a specific point.

The Derivative: Basic Definition

The derivative of a function \( f(x) \) is defined as:
\[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) – f(x)}{\Delta x} \]
This limit gives the slope of the tangent line to \( f(x) \) at point \( x \).

Basic Differentiation Rules

Power Rule

If \( f(x) = x^n \), then:
\[ f'(x) = n \cdot x^{n-1} \]
Example:
\[ \text{If } f(x) = x^3, \text{ then } f'(x) = 3x^2 \]

Constant Rule

If \( f(x) = c \) (where \( c \) is constant):
\[ f'(x) = 0 \]

Sum and Difference Rule

For \( f(x) = g(x) \pm h(x) \):
\[ f'(x) = g'(x) \pm h'(x) \]

Product Rule

For \( f(x) = u(x) \cdot v(x) \):
\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]

Quotient Rule

For \( f(x) = \frac{u(x)}{v(x)} \):
\[ f'(x) = \frac{v(x)u'(x) – u(x)v'(x)}{[v(x)]^2} \]

Applications of Differentiation

Finding Tangents to Curves

The derivative gives the slope of the tangent line at any point on a curve.

Example: Find the tangent to \( y = x^2 \) at \( x = 3 \).

1. Differentiate: \( y’ = 2x \)
2. Evaluate at \( x = 3 \): \( y’ = 6 \)
3. Equation: \( y – 9 = 6(x – 3) \) or \( y = 6x – 9 \)

Velocity and Acceleration

For position \( s(t) \):

• Velocity: \( v(t) = \frac{ds}{dt} \)
• Acceleration: \( a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2} \)

Optimization Problems

Critical points (maxima/minima) occur where \( f'(x) = 0 \).

Example Problem

Problem: Find critical points of \( f(x) = 3x^4 – 4x^3 + 2x^2 – 5 \).

Solution:

1. Differentiate:
   \[ f'(x) = 12x^3 – 12x^2 + 4x \]
2. Set \( f'(x) = 0 \):
   \[ 4x(3x^2 – 3x + 1) = 0 \]
3. Only real solution: \( x = 0 \)

Common Mistakes

1. Misapplying the Power Rule (e.g., forgetting to subtract 1 from the exponent).
2. Confusing Product Rule with Sum Rule.
3. Incorrectly applying the Quotient Rule numerator order.

Practice Questions

1. Differentiate \( f(x) = 5x^3 + 2x^2 – x + 7 \).
2. Find the tangent to \( y = x^3 – 4x^2 + 2x \) at \( x = 2 \).
3. Given \( s(t) = t^2 – 4t + 5 \), find velocity at \( t = 3 \).