Comprehensive Calculus Notes: Differentiation Rules, Applications, and Chain Rule

Limit Definition of the Derivative and the Power Law

• Limit Definition Mechanics:

  • Evaluating derivatives using the limit definition involves setting up the expression \n  \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}\n  
  • Expanding $f(x+h)$ and combining like terms leads to term cancellations in the numerator.
  • Both terms remaining in the numerator typically contain an $h$, allowing $h$ to be factored out.
  • Factoring allows the $h$ in the numerator to cancel the $h$ in the denominator.
  • Canceling $h$ resolves the indeterminate form of dividing by zero, which is mathematically and physically invalid.
  • Example evaluation step for $f(x) = x^2$:
  • Substituting into the definition and simplifying yields $2x + h$
  • Applying the limit as $h \rightarrow 0$ sets $h = 0$, yielding the derivative $2x$

• Derivation Pattern and the Power Law:

  • Evaluating standard monomial functions demonstrates a clear pattern:
  • \n    \frac{d}{dx}[x] = 1\n    
  • \n    \frac{d}{dx}[x^2] = 2x\n    
  • \n    \frac{d}{dx}[x^3] = 3x^2\n    
  • \n    \frac{d}{dx}[x^4] = 4x^3\n    
  • This pattern continues for higher powers such as $x^{10}$.
  • The Power Law General Rule: For any exponent $n$, the exponent comes down in front to multiply the variable, and the original exponent is reduced by $1$:     \n    \frac{d}{dx}[x^n] = n x^{n-1}\n    

Basic Differentiation Rules: Constant, Constant Multiple, Sum, and Difference Rules

• Constant Function Rule:

  • Any base raised to the power $0$ equals $1$ ($x^0 = 1$).
  • A constant function $f(x) = C$ (e.g., $f(x) = 6$, $f(x) = 1000$, or $f(x) = 10^{12}$) yields the exact same output for all inputs to infinity.
  • Evaluating a constant function using the limit definition:
  • $f(x+h) = 6$
  • Difference quotient: \n    \frac{6 - 6}{h} = \frac{0}{h} = 0\n    
  • Applying the limit is unnecessary as the expression simplifies to $0$ directly.
  • Rule: The derivative of any constant function is zero:     \n    \frac{d}{dx}[C] = 0\n    
  • Adding a constant to a function (e.g., $f(x) + 1$) leaves the derivative of the primary function unchanged because the constant term differentiates to $0$

• Constant Multiple Rule:

  • If a function is multiplied by a constant $c$, the constant can be factored outside of the differentiation operation:     \n    \frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)]\n    
  • Applies to known numerical constants or constant parameters representing physical quantities (such as constant pressure, velocity, or volume $V$).

• Sum and Difference Rules:

  • Sum Rule: The derivative of a sum of two functions equals the sum of their individual derivatives (distributive property of differentiation):     \n    \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]\n    
  • Difference Rule: Subtraction is the addition of a negative function; differentiation distributes across subtraction identically:     \n    \frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}[f(x)] - \frac{d}{dx}[g(x)]\n    

Real-World Application: Fluid Dynamics and Poiseuille's Law

• Physical Model of Fluid Flow in a Cylindrical Tube:

  • Models the flow velocity $v$ of a viscous fluid (such as blood containing white blood cells, or water in a pipe) moving through a cylindrical vessel or tube.
  • Parameters and Variables:
  • $R$ = Total outer radius of the cylindrical vessel (constant).
  • $r$ = Distance from the central axis of the tube ($0 \le r \le R$, variable).
  • $P$ = Pressure difference between the two ends of the tube (constant).
  • $\eta$ = Fluid viscosity, representing internal friction and resistance as blood cells rub against vessel walls (constant).
  • $l$ = Length of the vessel section (constant).
  • Velocity distribution: Fluid travels fastest along the central axis ($r = 0$) and slows down closer to the vessel walls ($r = R$).

• Mathematical Simplification and Differentiation:

  • Velocity equation with grouped constant $C$:     \n    v(r) = C(R^2 - r^2) = C R^2 - C r^2\n         where $C = \frac{P}{4 \eta l}$.
  • Differentiating velocity $v(r)$ with respect to radial position $r$ ($\frac{dv}{dr}$):
  • Apply Sum/Difference and Constant Multiple Rules:       \n      \frac{dv}{dr} = \frac{d}{dr}[C R^2] - \frac{d}{dr}[C r^2]\n      
  • Since $C R^2$ contains no variable $r$, it is purely constant, so $\frac{d}{dr}[C R^2] = 0$.
  • Applying the power rule to $C r^2$ gives:       \n      \frac{d}{dr}[C r^2] = C \cdot (2r) = 2 C r\n      
  • Substituting back the original constant parameters and simplifying fractions yields the rate of change of velocity across the radius:       \n      \frac{dv}{dr} = -2 C r = -\frac{P r}{2 \eta l}\n      

Differentiation of Exponential Functions

• Natural Exponential Function:

  • The derivative of $e^x$ is itself:     \n    \frac{d}{dx}[e^x] = e^x\n    
  • Proof using the limit definition:
  • $f(x+h) = e^{x+h} = e^x \cdot e^h$
  • Expression: \n    \lim_{h \rightarrow 0} \frac{e^x e^h - e^x}{h} = \lim_{h \rightarrow 0} \left( e^x \cdot \frac{e^h - 1}{h} \right)\n    
  • Factoring $e^x$ outside the limit (since it does not depend on $h$):       \n      e^x \cdot \lim_{h \rightarrow 0} \frac{e^h - 1}{h}\n      
  • Using the exponential limit identity $\lim_{h \rightarrow 0} \frac{e^h - 1}{h} = 1$, the limit simplifies to:       \n      e^x \cdot 1 = e^x\n      

• General Exponential Rules:

  • Exponential with constant multiplier in exponent $a$:     \n    \frac{d}{dx}[e^{ax}] = a e^{ax}\n    
  • Example with $a = -1$:     \n    \frac{d}{dt}[e^{-t}] = -e^{-t}\n    
  • General constant base $a$ (where $a > 0$):     \n    \frac{d}{dx}[a^x] = a^x \ln(a)\n    

• Worked Example:

  • Function: $f(x) = e^x - x^4$
  • Distribute derivative: \n  \frac{d}{dx}[f(x)] = \frac{d}{dx}[e^x] - \frac{d}{dx}[x^4]\n  
  • Result: \n  f'(x) = e^x - 4x^3\n  

Differentiation of Trigonometric Functions

• Basic Trigonometric Derivatives:

  • Derivative of Sine:     \n    \frac{d}{dx}[\sin(x)] = \cos(x)\n    
  • Derivative of Cosine:     \n    \frac{d}{dx}[\cos(x)] = -\sin(x)\n    

• Graphical Interpretation of Derivatives:

  • Tangent lines drawn along the curve $y = \sin(x)$ represent local slopes.
  • At peak and trough points of $y = \sin(x)$, horizontal tangent lines have a slope of $0$.
  • Plotting these slope values across all $x$ produces the exact curve of $y = \cos(x)$.
  • Performing the reverse process on $y = \cos(x)$ by mapping slope values yields the curve of $y = -\sin(x)$.

Product Rule for Differentiation

• Rule Definition and Formula:

  • Differentiation cannot be performed by simply multiplying the individual derivatives of two multiplied functions:     \n    \frac{d}{dx}[f(x) \cdot g(x)] \neq f'(x) \cdot g'(x)\n    
  • Gottfried Wilhelm Leibniz developed the correct product rule.
  • Product Rule Formula:     \n    \frac{d}{dx}[f(x) \cdot g(x)] = f(x) g'(x) + g(x) f'(x)\n    
  • Shorthand: $(fg)' = f g' + g f'$

• Worked Example:

  • Function: $h(x) = x^2 \sin(x)$
  • Identify sub-functions:
  • $f(x) = x^2 \implies f'(x) = 2x$
  • $g(x) = \sin(x) \implies g'(x) = \cos(x)$
  • Apply formula:     \n    h'(x) = x^2 \cdot \frac{d}{dx}[\sin(x)] + \sin(x) \cdot \frac{d}{dx}[x^2] = x^2 \cos(x) + 2x \sin(x)\n    

Quotient Rule for Differentiation

• Rule Definition and Formula:

  • Used for differentiating one function divided by another.
  • Formula:     \n    \frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{v(x) u'(x) - u(x) v'(x)}{[v(x)]^2}\n    
  • Alternative formulation using Product Rule and negative exponents:     \n    \frac{u(x)}{v(x)} = u(x) \cdot [v(x)]^{-1}\n    

• Worked Example:

  • Function: $w(x) = \frac{x^2 + x - 2}{x^3 + 6}$
  • Define numerator and denominator:
  • $u(x) = x^2 + x - 2 \implies u'(x) = 2x + 1$
  • $v(x) = x^3 + 6 \implies v'(x) = 3x^2$
  • Set up Quotient Rule:     \n    w'(x) = \frac{(x^3 + 6)(2x + 1) - (x^2 + x - 2)(3x^2)}{(x^3 + 6)^2}\n    
  • Expand numerator terms:
  • $(x^3 + 6)(2x + 1) = 2x^4 + x^3 + 12x + 6$
  • $(x^2 + x - 2)(3x^2) = 3x^4 + 3x^3 - 6x^2$
  • Combine numerator:     \n    (2x^4 + x^3 + 12x + 6) - (3x^4 + 3x^3 - 6x^2) = -x^4 - 2x^3 + 6x^2 + 12x + 6\n    
  • Final simplified derivative:     \n    w'(x) = \frac{-x^4 - 2x^3 + 6x^2 + 12x + 6}{(x^3 + 6)^2}\n    

Questions and Discussion on Homework and Exam Requirements

• Use of Limit Definition vs. Differentiation Rules:

  • Announcement made regarding Section 3.1 homework problems.
  • Even if standard differentiation rules are already known from high school, assigned problems in Section 3.1 specifically require showing full work using the limit definition of the derivative.
  • On exams, writing out the limit definition is not required unless explicitly specified; equation sheets will be provided.

• Homework Grading Standards:

  • Work Requirement: Submitting only a final answer results in zero credit. Homework requires showing logical sequences of mathematical steps to demonstrate process understanding.
  • Explanations: Annotating steps (e.g., explicitly stating "used product rule here") is highly recommended.
  • Formatting: Answers must be stated as proper logical equations (e.g., $y = \dots$) and include appropriate physical units where applicable.

The Chain Rule for Composite Functions

• Composite (Compound) Functions:

  • Functions nested within other functions, conceptualized as a "parent/mother function" containing a "baby function".
  • Examples:
  • $\sqrt{x^4 + 1}$: Outer parent function is $\sqrt{u}$, inner baby function is $u = x^4 + 1$
  • $e^{x^4}$: Outer parent function is $e^u$, inner baby function is $u = x^4$
  • Nested functions can have multiple layers (a baby function containing another baby function).

• Chain Rule Definition:

  • For a composite function $f(g(x))$:     \n    \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\n    
  • Procedure: Differentiate the outer parent function while leaving the inner baby function untouched inside, then multiply by the derivative of the inner baby function.
  • Leibniz notation:     \n    \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\n    

• Worked Example:

  • Function: $y = \sin(x^2)$
  • Parent function: $\sin(u)$
  • Baby function: $u = x^2$
  • Differentiate parent function while leaving baby inside: $\cos(x^2)$
  • Differentiate baby function: $\frac{d}{dx}[x^2] = 2x$
  • Multiply results together:     \n    \frac{dy}{dx} = 2x \cos(x^2)\n