Differentiation: Limits, Derivatives, and Applications in Calculus

Limits and the Definition of the Derivative

Difference Quotient and Slope of a Curve

The difference quotient is a fundamental concept in calculus used to approximate the slope of a curve at a given point. The slope at a point x=a is given by the limit:

• Definition: $\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Application: Used to find the instantaneous rate of change (derivative) at a specific point.

• Example: For $y = x^2 + 9x + 8$ at $x = 1$:

  • $f(1) = 1^2 + 9 \cdot 1 + 8 = 18$

  • $f(1+h) = (1+h)^2 + 9(1+h) + 8 = 1 + 2h + h^2 + 9 + 9h + 8 = 18 + 11h + h^2$

  • Difference quotient: $\frac{18 + 11h + h^2 - 18}{h} = \frac{11h + h^2}{h} = 11 + h$

  • Limit as $h \to 0$: $11$

Limit Definition of the Derivative

The derivative of a function at a point is defined as:

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

• Example: For $f(x) = 5x + 9$, $f'(2) = \lim_{h \to 0} \frac{5(2+h) + 9 - (5 \cdot 2 + 9)}{h} = \lim_{h \to 0} \frac{10 + 5h + 9 - 10 - 9}{h} = \lim_{h \to 0} \frac{5h}{h} = 5$

Continuity and Differentiability

Right-Hand and Left-Hand Derivatives

A function is differentiable at a point if the right-hand and left-hand derivatives are equal at that point. If they differ, the function is not differentiable there.

• Example: For a piecewise function:

  • Left-hand slope at $x=0$: $1$

  • Right-hand slope at $x=0$: $2$

  • Conclusion: Not differentiable at $x=0$

• Continuity: A function must be continuous at a point to be differentiable there, but continuity alone does not guarantee differentiability.

Basic Derivative Rules

Power Rule

• Formula: $\frac{d}{dx} x^n = n x^{n-1}$

• Example: $y = 11 - 4x^2$; $y' = -8x$

Sum and Difference Rules

• Formula: $\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)$

• Example: $y = 2x^2 + 8x + 2x^3$; $y' = 4x + 8 + 6x^2$

Product Rule

• Formula: $\frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)$

• Example: $y = (x-1)(3x-1)$; $y' = (x-1) \cdot 3 + (3x-1) \cdot 1 = 3(x-1) + (3x-1)$

Quotient Rule

• Formula: $\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

• Example: $y = \frac{x^2 - 2x}{x - 2}$; $y' = \frac{(2x - 2)(x - 2) - (x^2 - 2x)(1)}{(x - 2)^2}$

Applications of Derivatives

Tangent Lines

• Equation of Tangent: $y - y_1 = m(x - x_1)$, where $m = f'(x_1)$

• Example: Find the tangent to $y = \frac{18}{x^2 + 2}$ at $(1,6)$:

  • Find $y'$ at $x=1$

  • Plug into tangent line formula

Motion: Velocity and Acceleration

• Displacement: $s(t)$

• Velocity: $v(t) = s'(t)$

• Acceleration: $a(t) = v'(t) = s''(t)$

• Example: For $s = 2t^2 + 2t + 4$, $v = 4t + 2$, $a = 4$

Related Rates

• Definition: Problems where two or more related quantities change with respect to time.

• Example: The area $A$ of a circle with radius $r$ changes as $r$ changes: $A = \pi r^2$; $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$

Table: Summary of Derivative Rules

Rule	Formula	Example
Power Rule	$\frac{d}{dx} x^n = n x^{n-1}$	$\frac{d}{dx} x^3 = 3x^2$
Sum Rule	$\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$	$\frac{d}{dx} (x^2 + x) = 2x + 1$
Product Rule	$\frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)$	$\frac{d}{dx} (x^2 \cdot x) = 2x \cdot x + x^2 \cdot 1 = 3x^2$
Quotient Rule	$\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$	$\frac{d}{dx} \left( \frac{x^2}{x} \right) = \frac{2x \cdot x - x^2 \cdot 1}{x^2} = \frac{x^2}{x^2} = 1$

Additional info:

• Some problems involve evaluating derivatives at specific points, using both the limit definition and shortcut rules.

• Piecewise functions require checking both continuity and differentiability at the joining point.

• Motion problems use derivatives to find velocity and acceleration, and related rates problems apply the chain rule to connect changing quantities.