Unit 2: Differential calculus and its applications

I. Orientation — The derivative as a rate of change

Differential calculus studies how a function changes locally. For (y=f(x)), the derivative is the limiting ratio of the change in (y) to the corresponding change in (x).

• Definition:
  
  TEXT

    f'(x) = dy/dx = lim[h→0] {f(x+h) − f(x)}/h

  
  Here, (x) is the independent variable, (y) is the dependent variable, (h) is an increment in (x), and (f'(x)) is the derivative.
• Geometrical meaning: (f'(a)) is the slope of the tangent to (y=f(x)) at (x=a).
• Physical meaning: If (s(t)) is displacement at time (t), then (ds/dt) is velocity and (d^2s/dt^2) is acceleration.
• Differentiability condition: A function must have a finite, unique derivative at the point. Differentiability implies continuity, but continuity does not necessarily imply differentiability.
• Notation: Common forms are (f'(x)), (y'), (dy/dx), and (D_xy).

II. Fundamental Differentiation — Standard results and operating rules

A. Derivatives of standard functions and general rules of differentiation

Standard derivatives and structural rules allow complicated functions to be differentiated without repeatedly applying the limit definition.

• Elementary functions:
  
  TEXT

    d(c)/dx = 0                    d(xⁿ)/dx = nxⁿ⁻¹
    d(eˣ)/dx = eˣ                 d(aˣ)/dx = aˣ ln a
    d(ln x)/dx = 1/x              d(logₐx)/dx = 1/(x ln a)
  
    d(sin x)/dx = cos x           d(cos x)/dx = −sin x
    d(tan x)/dx = sec²x           d(cot x)/dx = −csc²x
    d(sec x)/dx = sec x tan x     d(csc x)/dx = −csc x cot x

  
  Here, (c) is constant, (n) is real where the power is defined, and (a>0), (a\ne1).
• Inverse trigonometric results:
  
  TEXT

    d(sin⁻¹x)/dx = 1/√(1−x²)       d(cos⁻¹x)/dx = −1/√(1−x²)
    d(tan⁻¹x)/dx = 1/(1+x²)        d(cot⁻¹x)/dx = −1/(1+x²)

• Algebraic rules: For differentiable (u(x)), (v(x)), and constant (k),
  
  TEXT

    d(ku)/dx = k du/dx
    d(u±v)/dx = du/dx ± dv/dx
    d(uv)/dx = u dv/dx + v du/dx
    d(u/v)/dx = {v du/dx − u dv/dx}/v²,  v ≠ 0

• Chain rule: For (y=f(u)) and (u=g(x)),
  
  TEXT

    dy/dx = (dy/du)(du/dx)

  
  Thus, if (y=\sin(x^2)), then (dy/dx=2x\cos(x^2)).

III. Parametric Differentiation — Curves defined through a parameter

A. Derivatives of parametric forms

Parametric differentiation applies when both coordinates are expressed in terms of a third variable rather than directly relating (y) to (x).

• First derivative: If (x=f(t)) and (y=g(t)), then
  
  TEXT

    dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0

  
  Here, (t) is the parameter.
• Second derivative:
  
  TEXT

    d²y/dx² = {d/dt(dy/dx)}/(dx/dt)

• Example: For (x=t^2) and (y=t^3),
  
  TEXT

    dy/dx = 3t²/(2t) = 3t/2
    d²y/dx² = (3/2)/(2t) = 3/(4t)

  
  These expressions apply where (t\ne0).

IV. Implicit Differentiation — Relations not solved for one variable

A. Derivatives of implicit functions

Implicit differentiation finds derivatives when (x) and (y) satisfy an equation (F(x,y)=0).

• Principle: Differentiate every term with respect to (x), treating (y) as a function of (x).
• Chain-rule requirement:
  
  TEXT

    d[yⁿ]/dx = nyⁿ⁻¹ dy/dx

• General result: If the relevant partial derivatives exist and (F_y\ne0),
  
  TEXT

    dy/dx = −Fₓ/Fᵧ

  
  Here, (F_x) and (F_y) are partial derivatives of (F).
• Example: From (x^2+y^2=25),
  
  TEXT

    2x + 2y(dy/dx) = 0
    dy/dx = −x/y

V. Logarithmic Methods — Simplifying products and variable powers

A. Logarithmic differentiation

Logarithmic differentiation converts products into sums, quotients into differences, and powers into coefficients before differentiation.

• Method:
  1. Set the function equal to (y).
  2. Take natural logarithms on both sides.
  3. Apply logarithm laws.
  4. Differentiate implicitly and solve for (dy/dx).
• Variable base and exponent: For (y=u(x)^{v(x)}), with (u(x)>0),
  
  TEXT

    ln y = v ln u
    (1/y)dy/dx = v' ln u + v(u'/u)
    dy/dx = uᵛ[v' ln u + vu'/u]

  
  Here, (u'=du/dx) and (v'=dv/dx).
• Example: For (y=x^x), (x>0),
  
  TEXT

    dy/dx = xˣ(ln x + 1)

VI. Successive Differentiation — Rates of higher order

A. Higher order derivatives of simple functions

Higher-order derivatives are obtained by differentiating a function repeatedly.

• Notation:
  
  TEXT

    y'', y''', y⁽ⁿ⁾  or  d²y/dx², d³y/dx³, dⁿy/dxⁿ

  
  Here, (n) is the order of differentiation.
• Power function:
  
  TEXT

    dʳ(xⁿ)/dxʳ = n!xⁿ⁻ʳ/(n−r)!, for r ≤ n

  
  It becomes zero for integer (n\ge0) and (r>n).
• Exponential function:
  
  TEXT

    dⁿ(eᵃˣ)/dxⁿ = aⁿeᵃˣ

• Trigonometric functions:
  
  TEXT

    dⁿ(sin ax)/dxⁿ = aⁿ sin(ax+nπ/2)
    dⁿ(cos ax)/dxⁿ = aⁿ cos(ax+nπ/2)

  
  Here, (a) is constant.

VII. Rolle’s Theorem — Existence of a stationary point

A. Rolle's theorem

Rolle’s theorem guarantees at least one point with a horizontal tangent when a smooth curve has equal endpoint values.

• Conditions: A function (f) must be:
  • continuous on the closed interval ([a,b]);
  • differentiable on the open interval ((a,b));
  • equal at the endpoints, so (f(a)=f(b)).
• Conclusion:
  
  TEXT

    There exists c ∈ (a,b) such that f'(c) = 0.

  
  Here, (c) is at least one interior point.
• Geometrical meaning: Some tangent to the curve between (a) and (b) is parallel to the (x)-axis.
• Limitation: The theorem proves existence but does not identify (c) or guarantee that it is unique.

VIII. Mean-Value Principles — Connecting average and instantaneous change

A. Mean value theorems

Mean value theorems guarantee an interior point where a local derivative represents an average change over an interval.

• Lagrange’s mean value theorem: If (f) is continuous on ([a,b]) and differentiable on ((a,b)), then some (c\in(a,b)) satisfies
  
  TEXT

    f'(c) = {f(b)−f(a)}/(b−a)

  
  The tangent at (c) is parallel to the chord joining the endpoints.
• Rolle’s theorem as a special case: If (f(a)=f(b)), the average slope is zero and hence (f'(c)=0).
• Cauchy’s mean value theorem: If (f) and (g) are continuous on ([a,b]), differentiable on ((a,b)), and relevant denominators are nonzero, then
  
  TEXT

    f'(c)/g'(c) = {f(b)−f(a)}/{g(b)−g(a)}

  
  Here, (g) supplies a second scale of change.
• Consequence: If (f'(x)=0) throughout an interval, then (f) is constant there.

IX. Series Expansions — Local polynomial representation

A. Taylor's theorems

Taylor’s theorem represents a sufficiently differentiable function near (x=a) by a polynomial plus a remainder.

• Finite expansion:
  
  TEXT

    f(x) = f(a) + (x−a)f'(a) + (x−a)²f''(a)/2!
           + ⋯ + (x−a)ⁿf⁽ⁿ⁾(a)/n! + Rₙ

  
  Here, (n) is the polynomial degree and (R_n) is the remainder.
• Lagrange remainder:
  
  TEXT

    Rₙ = f⁽ⁿ⁺¹⁾(ξ)(x−a)ⁿ⁺¹/(n+1)!

  
  Here, (\xi) lies between (a) and (x).
• Purpose: The polynomial approximates nonlinear functions near (a); the remainder measures truncation error.
• Accuracy: A small (|x-a|) generally improves approximation when the higher derivative remains bounded.

B. Maclaurin theorems

Maclaurin’s theorem is Taylor’s theorem expanded about the origin, so (a=0).

• General form:
  
  TEXT

    f(x) = f(0) + xf'(0) + x²f''(0)/2! + ⋯ + xⁿf⁽ⁿ⁾(0)/n! + Rₙ

• Standard expansions:
  
  TEXT

    eˣ = 1 + x + x²/2! + x³/3! + ⋯
    sin x = x − x³/3! + x⁵/5! − ⋯
    cos x = 1 − x²/2! + x⁴/4! − ⋯
    ln(1+x) = x − x²/2 + x³/3 − ⋯,  −1 < x ≤ 1

• Engineering use: For small (x), (\sin x\approx x), (\cos x\approx1-x^2/2), and (e^x\approx1+x).

X. Indeterminate Limits — Forms requiring further analysis

A. Indeterminate forms

An indeterminate form is a limit form whose appearance alone does not determine the limit’s value.

• Basic forms:
  
  TEXT

    0/0,  ∞/∞,  0·∞,  ∞−∞,  0⁰,  1∞,  ∞⁰

• Why indeterminate: Different functions producing the same form may have finite, infinite, or nonexistent limits.
• Transformations:
  • Convert (0\cdot\infty) into a quotient.
  • Combine terms in (\infty-\infty) using a common denominator or rationalization.
  • For (y=f(x)^{g(x)}), use (\ln y=g(x)\ln f(x)) to handle power forms.
• Distinction: Forms such as (1/0) are undefined or divergent forms, not indeterminate forms.

XI. L’Hospital’s Method — Resolving quotient indeterminacy

A. L'Hospital's rule

L’Hospital’s rule evaluates suitable limits of the forms (0/0) or (\infty/\infty) by comparing derivatives.

• Rule: Under the theorem’s differentiability and denominator conditions,
  
  TEXT

    lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x)

  
  provided the derivative-ratio limit exists as a finite or infinite value.
• Repeated use: The rule may be applied again if the differentiated quotient remains (0/0) or (\infty/\infty).
• Example:
  
  TEXT

    lim[x→0] (eˣ−1)/x
    = lim[x→0] eˣ/1
    = 1

• Restriction: Products, differences, and powers must first be transformed into an appropriate quotient or logarithmic form.

XII. Optimization — Locating extreme values

A. Maxima and minima

Maxima and minima identify points where a function is locally or absolutely greatest or least.

• Critical points: Interior candidates occur where (f'(x)=0) or where (f'(x)) does not exist while (f(x)) is defined.
• First-derivative test:
  1. If (f') changes from positive to negative, (f) has a local maximum.
  2. If (f') changes from negative to positive, (f) has a local minimum.
• Second-derivative test: At a stationary point (x=c),
  
  TEXT

    f''(c) < 0  ⇒ local maximum
    f''(c) > 0  ⇒ local minimum
    f''(c) = 0  ⇒ test is inconclusive

• Absolute extrema: For a continuous function on ([a,b]), compare values at all critical points and at both endpoints.
• Engineering procedure: Define the objective function, express it using one independent variable, determine the feasible domain, locate critical points, and compare admissible values.
• Example: For (f(x)=x^2-4x+7),
  
  TEXT

    f'(x)=2x−4=0  ⇒  x=2
    f''(x)=2>0    ⇒  local minimum
    f(2)=3

  
  Thus, the minimum value is (3) at (x=2).