3.3.2 Rules of Differentiation

Slope of a Tangent Line (Finding where the derivative equals a prescribed value)

• Problem statement

  • Given $f(x)=2x^3-15x^2+24x$, find all $x$-values where the tangent line to the graph has slope $6$.
  • "Slope of the tangent" = value of the first derivative $f'(x)$ at the desired point.

• Differentiate with basic rules

  • Power Rule $(d/dx)[x^n]=nx^{n-1}$ and Constant Multiple Rule apply.
  • Compute
    \begin{aligned}
    f'(x) &= (d/dx)(2x^3) - (d/dx)(15x^2) + (d/dx)(24x)\
    &= 3\cdot 2x^{3-1} - 2\cdot 15x^{2-1} + 24\
    &= 6x^2 - 30x + 24.
    \end{aligned}

• Set derivative equal to target slope
  $6x^2 - 30x + 24 = 6.$

  • Subtract $6$ from both sides: $6x^2 - 30x + 18 = 0.$
  • Factor out common $6$: $6(x^2 - 5x + 3)=0 \Rightarrow x^2 - 5x + 3 = 0.$

• Quadratic Formula

  • General form $ax^2+bx+c=0 \Rightarrow x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}.$
  • Here $a=1,\;b=-5,\;c=3$:
    x = \frac{-(-5) \pm \sqrt{(-5)^2-4(1)(3)}}{2(1)}
    = \frac{5 \pm \sqrt{25-12}}{2}
    = \frac{5 \pm \sqrt{13}}{2}.
  • Decimal approximations (useful for graphing or intuition):
    $\frac{5-\sqrt{13}}{2} \approx 0.697, \qquad \frac{5+\sqrt{13}}{2} \approx 4.303.$

• Interpretation/Significance

  • At $x \approx 0.697$ and $x \approx 4.303$, the curve is rising with slope $6$.
  • In applications this process finds where a system meets a given rate-of-change requirement (e.g.
    economics: marginal cost = given value, physics: velocity equals specific value, etc.).

Higher-Order Derivatives (Concept & Notation)

• Idea

  • "Higher order" means differentiating repeatedly.
  • Notation:
  • First derivative: $f'(x)$ or $\frac{dy}{dx}.$
  • Second derivative: $f''(x) = \frac{d^2y}{dx^2}$ — captures concavity/acceleration.
  • Third derivative: $f'''(x) = \frac{d^3y}{dx^3}$ — appears in physics as "jerk" (rate of change of acceleration).
  • $n$-th derivative: $f^{(n)}(x).$

• Why care?

  • Analyze curvature, inflection points, motion (position→velocity→acceleration→jerk→…).
  • Used in Taylor series, differential equations, control systems.

Example 1 — Polynomial (Find the third derivative)

• Reconstructed original function (deduced from derivative steps in the video):
  $g(x)=3x^3-5x+12.$

• Step-by-step differentiation

  • First derivative
    $g'(x)=9x^2-5$ (since $(d/dx)(3x^3)=9x^2,$ $(d/dx)(-5x)=-5,$ constant $12$ vanishes).
  • Second derivative
    $g''(x)=(d/dx)(9x^2-5)=18x.$
  • Third derivative
    $g'''(x)=(d/dx)(18x)=18.$

• Observations

  • The third derivative of a cubic polynomial is always constant.
  • Physical analogy: a constant jerk implies linearly increasing acceleration.

Example 2 — Mixed Linear & Exponential (Find up to third derivative)

• Function
  $y(t)=3t+2e^{t}.$

• Notation reminder

  • Using Leibniz form $\frac{dy}{dt},\;\frac{d^2y}{dt^2},\;\frac{d^3y}{dt^3}.$

• Derivatives

  • First derivative
    $\frac{dy}{dt}=3+2e^{t},$ because $(d/dt)(3t)=3$ and $(d/dt)(e^{t})=e^{t}.$
  • Second derivative
    $\frac{d^2y}{dt^2}=2e^{t}$ (derivative of constant $3$ is $0$; $2e^{t}$ remains unchanged).
  • Third derivative
    $\frac{d^3y}{dt^3}=2e^{t}$ (exponential function differentiates to itself indefinitely).

• Pattern insight

  • For $ae^{t}$, every derivative is still $ae^{t}$.
  • Linear terms vanish after first differentiation.

Quick Reference: Rules Employed & Their Significance

• Power Rule: $\frac{d}{dx}(x^{n}) = nx^{n-1}$ (foundation for differentiating polynomials).
• Constant Multiple Rule: $\frac{d}{dx}[c\,f(x)] = c\,f'(x).$
• Sum/Difference Rule: differentiate term-by-term.
• Derivative of $e^{x}$: remains $e^{x}$ — makes exponentials pivotal in modeling continuous growth/decay.
• Quadratic Formula: universal method when factoring fails; discriminant $b^2-4ac$ signals number/nature of roots.

Broader Connections & Practical Takeaways

• Tangent‐slope problems illustrate how derivatives convert geometric questions into algebraic ones.
• Higher-order derivatives underpin concavity testing, optimization (via second derivative test), and series approximations (Taylor/Maclaurin).
• Exponentials’ self-replicating derivatives explain their ubiquity in differential equations (e.g.
  Newton’s law of cooling, population models).
• Recognizing when to switch from factoring to quadratic formula saves time and prevents errors.

Suggested Practice / Next Steps

• Re-work the tangent‐slope example with a different target slope (e.g.
  $m=10$) to reinforce the derivative-to-algebra transition.
• Compute 4th and 5th derivatives of the exponential example to verify the pattern.
• Sketch $f(x)=2x^3-15x^2+24x$ and mark the points where the slope equals $6$ to internalize the geometric meaning.
• Read ahead on the second derivative test for local maxima/minima to see higher-order derivatives in action.