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 $P(x<em>0,f(x</em>0))$ and $Q(x<em>1,f(x</em>1))$.
  • Slope of secant: $m<em>{\overleftrightarrow{PQ}} = \dfrac{f(x</em>1)-f(x<em>0)}{x</em>1-x<em>0}=\dfrac{f(x</em>0+\Delta x)-f(x<em>0)}{\Delta x}$ where $\Delta x=x</em>1-x_0$.
• 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 $P$:
    $m<em>{\ell}=\lim</em>{\Delta x\to 0}\dfrac{f(x<em>0+\Delta x)-f(x</em>0)}{\Delta x}=\lim<em>{x</em>1\to x<em>0}\dfrac{f(x</em>1)-f(x<em>0)}{x</em>1-x_0}.$
  • 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 $m<em>{NL}=-1/m</em>{TL}$ 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 $x<em>0$: $f'(x</em>0)=\lim<em>{x\to x</em>0}\dfrac{f(x)-f(x<em>0)}{x-x</em>0}.$
• 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: $D<em>x(5)=0,\;D</em>x[x^5]=5x^4,\;D_x\left[\dfrac1{x^2}\right]=-\dfrac{2}{x^3}.$
• Constant multiple: $D<em>x[3x^2]=6x,\;D</em>x[(2x)^5]=160x^4$ (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: $\lim<em>{h\to0}\dfrac{\sin h}{h}=1$ and $\lim</em>{h\to 0}\dfrac{1-\cos h}{h}=0.$
• 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.