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.