Slopes, Derivatives & Their Rules – Comprehensive Notes

The Tangent Line

• Historical intuition: for a circle, the tangent at point P touches the circle but does not cut through; we generalize this idea to any curve.
• Secant line construction
  • Two points on the curve P(x0,f(x0)) and Q(x1,f(x1)).
  • Slope of secant: m{\overleftrightarrow{PQ}} = \dfrac{f(x1)-f(x0)}{x1-x0}=\dfrac{f(x0+\Delta x)-f(x0)}{\Delta x} where \Delta x=x1-x_0.
• Tangent line obtained by letting Q \to P \,(\Delta x \to 0) so that the secant approaches a unique limiting line \ell.
  • Slope of tangent at P:
    m{\ell}=\lim{\Delta x\to 0}\dfrac{f(x0+\Delta x)-f(x0)}{\Delta x}=\lim{x1\to x0}\dfrac{f(x1)-f(x0)}{x1-x_0}.
  • If the above limit exists, \ell is called the tangent line at P.
  • If the one-sided limits blow up to \pm\infty the vertical line x=x_0 is the tangent. Otherwise no tangent exists.
• Geometric/qualitative remarks
  • Magnitude of m_{\ell} gauges flatness vs. steepness.
  • Sign of m_{\ell} tells whether the curve rises (positive) or falls (negative) at P.
  • A tangent line may intersect the curve again away from the point of tangency.
• Normal line: the line through P perpendicular to the tangent; slope m{NL}=-1/m{TL} whenever m_{TL}\neq 0.

Worked Example – Tangent & Normal to f(x)=\dfrac1x at x=1

1. Point: (1,1).
2. Tangent slope via limit:
   \begin{aligned}
   m{TL}&=\lim{\Delta x\to 0}\frac{\tfrac1{1+\Delta x}-1}{\Delta x}
   =\lim{\Delta x\to 0}\frac{1-(1+\Delta x)}{\Delta x(1+\Delta x)} =\lim{\Delta x\to 0}\frac{-\Delta x}{\Delta x(1+\Delta x)}=-1.
   \end{aligned}
3. Tangent line (point–slope): y-1=-\,(x-1).
4. Normal slope =+1, normal line: y-1=(x-1).
5. General slope for the same function: m_{TL}(x)=-\dfrac1{x^2} (demonstrates a tangible derivative function).

Definition of the Derivative

• Derivative of f is the function
  f'(x)=\lim_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}.
• Domain restrictions: derivative may fail to exist at some points ⇒ \text{dom}(f')\subseteq\text{dom}(f).
• Alternative notation at a single point x0: f'(x0)=\lim{x\to x0}\dfrac{f(x)-f(x0)}{x-x0}.
• Synonyms/notations: y'\,,\;\dfrac{dy}{dx}\,,\;D_x[f(x)]\,,\;\dfrac{d}{dx}[f(x)]. The computational process is called differentiation.

Example via Definition

For f(x)=\sqrt{x}:
\begin{aligned}
f'(x)&=\lim{\Delta x\to 0}\frac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x} \cdot\frac{\sqrt{x+\Delta x}+\sqrt{x}}{\sqrt{x+\Delta x}+\sqrt{x}} =\lim{\Delta x\to 0}\frac{(x+\Delta x)-x}{\Delta x(\sqrt{x+\Delta x}+\sqrt{x})}
=\frac1{2\sqrt{x}}.
\end{aligned}

Differentiation Rules

• Constant Rule: \dfrac{d}{dx}[c]=0.
• Power Rule (rational exponent n): \dfrac{d}{dx}[x^n]=n x^{n-1}.
• Constant Multiple: \dfrac{d}{dx}[c\,g(x)]=c\,g'(x).
• Sum / Difference: \dfrac{d}{dx}[f\pm g]=f'\pm g'.
• Product Rule: (fg)'=f'g+fg'.
• Quotient Rule: \left(\dfrac{f}{g}\right)'=\dfrac{g f'-f g'}{g^{2}}, \;g\neq 0.

Illustrative Computations

• Constant & power: Dx(5)=0,\;Dx[x^5]=5x^4,\;D_x\left[\dfrac1{x^2}\right]=-\dfrac{2}{x^3}.
• Constant multiple: Dx[3x^2]=6x,\;Dx[(2x)^5]=160x^4 (emphasizes expansion before power rule).
• Sum/Difference: D_x[x^2+3x]=2x+3, \dfrac{d}{dx}\bigl(2x^4-\tfrac{4}{5}\sqrt{x}+7\bigr)=8x^3-\dfrac{2}{5\sqrt{x}}.
• Product: \dfrac{d}{dx}[(x-3)(2x^2-3)]=6x^2-12x-3.
• Quotient (template example): for f(x)=\dfrac{2x^3-1}{x^3+4}\Big/ (3x-5) notation reminder given.

Derivatives of Trigonometric Functions

• Fundamental list (proof of \sin given via limit laws):
  1. \dfrac{d}{dx}[\sin x]=\cos x
  2. \dfrac{d}{dx}[\cos x]=-\sin x
  3. \dfrac{d}{dx}[\tan x]=\sec^2 x
  4. \dfrac{d}{dx}[\cot x]=-\csc^2 x
  5. \dfrac{d}{dx}[\sec x]=\sec x\tan x
  6. \dfrac{d}{dx}[\csc x]=-\csc x\cot x
• Mixed examples
  • \dfrac{d}{dx}[3\sin x-7\cos x]=3\cos x+7\sin x.
  • Product of trigs: D_x[\sec x\,\csc x]=(\sec x\tan x)(\csc x)+\sec x(-\csc x\cot x).
  • Combined quotient: If f(x)=\dfrac{\cot x - x}{1+\tan x} then
    f'(x)=\dfrac{(1+\tan x)(-\csc^{2}x-1)-(\cot x - x)\sec^{2}x}{(1+\tan x)^2}.

Higher-Order Derivatives

• Second derivative: derivative of f', denoted f''.
• n^{\text{th}} derivative recursively defined by
  f^{(n)}(x)=\lim_{\Delta x\to 0}\dfrac{f^{(n-1)}(x+\Delta x)-f^{(n-1)}(x)}{\Delta x}.
• Notational variants: y^{(n)},\;\dfrac{d^n y}{dx^n},\;D_x^{\,n}[f(x)].
• Order terminology: n is the order; f=f^{(0)}.

Polynomial Example

For f(x)=x^6-x^4-3x^3+2x^2-4 successive derivatives are
\begin{aligned}
f'(x)&=6x^5-4x^3-9x^2+4x,\
f''(x)&=30x^4-12x^2-18x+4,\
f'''(x)&=120x^3-24x-18,\
f^{(4)}(x)&=360x^2-24,\
f^{(5)}(x)&=720x,\
f^{(6)}(x)&=720,\
f^{(n)}(x)&=0\;\text{for all}\;n\ge 7.
\end{aligned}

• Observation: for a degree-m polynomial the derivative becomes identically zero after m steps.

Conceptual Connections & Significance

• Tangent slope = instantaneous rate of change; foundation for velocities, growth rates, marginal analysis in economics.
• Normal lines arise in physics (reflection/refraction), computer graphics (lighting normals).
• Differentiation rules accelerate computation, replacing repeated limit evaluations.
• Trigonometric derivative identities underpin solutions of oscillatory models (waves, circuits, optics).
• Higher derivatives relate to curvature, Taylor series, differential equations, and physical quantities such as jerk (third derivative of position).
• Vertical tangents signal possible cusp/critical behavior important in optimization and qualitative sketching.

Typical Exam-Style Exercises (from transcript)

1. Tangent & normal to f(x)=2x\cos x at x=\pi.
2. Compute \dfrac{dy}{dx}\, for y=2x\cos x (shortcut via product rule).
3. Differentiate the composite expression x-8\sqrt[4]{x}\sec x-3\cot x.
4. Find f^{(17)}(x) for f(x)=\sin x (exploit periodic cycling of sine derivatives every 4 steps).

Study Tips

• Always start with recognition: constant, power, trig, product, quotient.
• For limits, memorize fundamental trig limits: \lim{h\to0}\dfrac{\sin h}{h}=1 and \lim{h\to 0}\dfrac{1-\cos h}{h}=0.
• Check domain issues (e.g.
  cusps, vertical tangents) before concluding differentiability.
• Organize multi-rule problems hierarchically: outermost rule → inner rules.
• For high-order derivatives of trig / exponentials, note pattern cycles.

Ethical / Practical Remarks

• Calculus models real-world change; misuse (e.g.
  ignoring domain constraints) can lead to erroneous engineering or economic predictions.
• Clear communication of derivatives (notation, units) is essential for interdisciplinary collaboration.

Quick Reference – Formula Sheet

• Constant: 0 | Power: nx^{n-1} | c\,f' | f'\pm g' | (fg)'=f'g+fg' | (f/g)'=(gf'-fg')/g^2
• Trig: \cos, -\sin, \sec^2, -\csc^2, \sec\tan, -\csc\cot
• Higher derivatives: apply rules repeatedly; polynomials terminate; \sin/\cos cycle every 4.