3.3.1 Rules of Differentiation

Motivation for Differentiation Rules

• Up to now, derivatives were computed directly from the definition (limits) – correct but time-consuming.
• A collection of rules streamlines differentiation, making calculus practical for everyday and advanced work.
• Knowing the rules well is “the good stuff” that students keep using in later courses.

Constant Rule

• Statement: If c is any real number, then \frac{d}{dx}(c)=0.
  • Geometric reasoning: y=c is a horizontal line; slope at every point is 0.
• Example
  • \frac{d}{dx}(2^8)=0 (even though 2^8=256 looks large, it’s still just a constant).

Power Rule (MOST-USED)

• Let n be a non-negative integer.
  \frac{d}{dx}(x^{n}) = n x^{n-1}
• Procedure: “Pull the exponent down, subtract 1 from it.”
• The text gives a formal proof (worth reading if curious).
• Examples
  • \frac{d}{dx}(x^9)=9x^8.
  • \frac{d}{dx}(x)=\frac{d}{dx}(x^1)=1\cdot x^{0}=1 (agrees with the fact that the line y=x has slope 1).

Constant Multiple Rule

• If c is a constant and f differentiable, then
  \frac{d}{dx}\bigl(c\,f(x)\bigr)=c\,f'(x).
• In practice: drag the constant outside → apply other rules to the variable part.
• Example
  • f(x)=-\tfrac78 x^{11}
    f'(x)=-\tfrac78\cdot 11 x^{10}=-\tfrac{77}{8}x^{10}.

Rewriting Roots for the Power Rule

• Square root \sqrt{t}=t^{1/2}, etc.
• Example
  • g(t)=\tfrac38\sqrt{t}=\tfrac38 t^{1/2}
    g'(t)=\tfrac38\cdot\tfrac12 t^{-1/2}=\tfrac{3}{16}t^{-1/2}=\tfrac{3}{16\sqrt{t}}.

Sum (and Difference) Rule

• If f and g are differentiable,
  \frac{d}{dx}\bigl(f(x)\pm g(x)\bigr)=f'(x)\pm g'(x) (same sign).
• Justified because derivatives are limits, and limit laws allow term-by-term work.
• Example – Polynomial
  • h(w)=3w^3+9w^2+6w-8
  • Apply rules termwise:
    h'(w)=6w^2+18w+6 (constant −8 vanishes).

The Number e and Its Special Exponential Function

• e\approx2.7182818\dots is to exponentials what \pi is to circles.
• Characterized by \displaystyle\lim_{h\to0}\frac{e^{h}-1}{h}=1.
• Forms the base of the natural exponential e^{x}, whose inverse is the natural logarithm \ln x.

Unique Derivative Property

• \displaystyle \frac{d}{dx}(e^{x}) = e^{x} — the only elementary function equal to its own derivative.
• Proof outline is in the textbook; for now we employ the fact directly.

Worked Application: Tangent Line & Horizontal Tangents

Given f(x)=2x-\dfrac{e^{x}}{2} and the point (0,-\tfrac12).

Step 1 – Derivative

• Use difference, constant multiple, and e^{x} rules:
  f'(x)=\frac{d}{dx}(2x)-\frac{d}{dx}\Bigl(\tfrac12 e^{x}\Bigr)=2-\tfrac12 e^{x}.

Step 2 – Tangent Line at (0,-\tfrac12)

• Slope: m=f'(0)=2-\tfrac12 e^{0}=2-\tfrac12=\tfrac32.
• Point-slope form:
  y+\tfrac12=\tfrac32(x-0) → y=\tfrac32x-\tfrac12.

Step 3 – Horizontal Tangent(s)

• Horizontal means slope 0, so set derivative to zero:
  2-\tfrac12 e^{x}=0 \quad\Longrightarrow\quad e^{x}=4.
• Solve with natural log:
  x=\ln4.
• Corresponding point:
  \bigl(\,\ln4,\;f(\ln4)=2\ln4-\tfrac12 e^{\ln4}\bigr)=\bigl(\ln4,\;2\ln4-2\bigr).

Practical Tips & Connections

• Most derivative computations are combinations of the Constant, Power, Constant-Multiple, and Sum Rules.
• Rewriting radicals and reciprocals into power form is essential to leverage the Power Rule.
• The derivative rules mirror limit laws; understanding limits deepens your grasp of why these shortcuts work.
• e^{x}’s self-derivative property underlies continual growth models, compound interest, differential equations, and later, solving linear ODEs.
• Recognize that “horizontal tangent” questions translate to “solve f'(x)=0,” a key concept for optimization problems.