Unit 2: Differential calculus and its applications

I. Orientation: The Differential Calculus Framework

Differential calculus studies instantaneous change through the derivative, developed systematically by Isaac Newton and Gottfried Wilhelm Leibniz in the seventeenth century. For a function (y=f(x)), the derivative measures the limiting ratio of the change in (y) to the corresponding change in (x).

• Derivative definition: If the limit exists at (x), then
  

  TEXT

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

  
  Here, (h) is an increment in (x), (f'(x)) is the derivative, and (dy/dx) is Leibniz notation.

• Geometrical meaning: (f'(a)) is the slope of the tangent to (y=f(x)) at ((a,f(a))).

• Physical meaning: If displacement is (s(t)), then (ds/dt) is velocity and (d^2s/dt^2) is acceleration, where (t) denotes time.

• Differentiability condition: A function is differentiable at (x=a) only when its left and right derivatives exist, are finite, and are equal.

• Continuity relation: Differentiability at (a) implies continuity at (a), but continuity does not imply differentiability; (f(x)=|x|) is continuous but not differentiable at (x=0).

• Working convention: Angles in trigonometric differentiation are measured in radians; standard formulas such as (d(\sin x)/dx=\cos x) depend on this convention.

II. Fundamental Differentiation

A. Derivatives of standard functions and general rules of differentiation

Standard derivative formulas and combination rules make it possible to differentiate complex functions without repeatedly applying the limit definition.

• Algebraic and exponential functions:
  

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    d(c)/dx = 0                 d(x^n)/dx = nx^(n-1)
    d(e^x)/dx = e^x             d(a^x)/dx = a^x ln a
    d(ln x)/dx = 1/x            d(log_a x)/dx = 1/(x ln a)

  
  Here, (c) is constant, (n) is real where defined, (a>0), (a\ne1), and (\ln) is the natural logarithm.

• Trigonometric functions:
  

  TEXT

    d(sin x)/dx = cos x         d(cos x)/dx = -sin x
    d(tan x)/dx = sec^2 x       d(cot x)/dx = -csc^2 x
    d(sec x)/dx = sec x tan x   d(csc x)/dx = -csc x cot x

• Inverse trigonometric functions:
  

  TEXT

    d(sin^(-1)x)/dx = 1/sqrt(1-x^2)
    d(cos^(-1)x)/dx = -1/sqrt(1-x^2)
    d(tan^(-1)x)/dx = 1/(1+x^2)

• General rules: For differentiable (u(x)) and (v(x)),
  

  TEXT

    d(ku)/dx = k du/dx
    d(u±v)/dx = u'±v'
    d(uv)/dx = u'v+uv'
    d(u/v)/dx = (vu'-uv')/v^2
    d[f(g(x))]/dx = f'(g(x))g'(x)

  
  Here, (k) is constant, (v\ne0), and the last formula is the chain rule.

• Worked example: For (y=x^2e^x), the product rule gives
  

  TEXT

    dy/dx = 2xe^x + x^2e^x = e^x(x^2+2x).

III. Parametric Differentiation

A. Derivatives of parametric forms

When (x) and (y) are expressed through a parameter, their derivative is obtained by eliminating the parameter’s rate of change.

• First derivative: If (x=f(t)), (y=g(t)), and (dx/dt\ne0), then
  

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    dy/dx = (dy/dt)/(dx/dt).

  
  Here, (t) is the parameter.

• Second derivative:
  

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    d^2y/dx^2 = {d/dt(dy/dx)}/(dx/dt).

  
  Differentiating (dy/dx) directly with respect to (t) alone does not produce (d^2y/dx^2).

• Worked example: For (x=t^2) and (y=t^3),
  

  TEXT

    dy/dx = 3t^2/(2t) = 3t/2,
    d^2y/dx^2 = (3/2)/(2t) = 3/(4t).

IV. Implicit Differentiation

A. Derivatives of implicit functions

Implicit differentiation applies when an equation relates (x) and (y) without isolating (y).

• Principle: Differentiate both sides with respect to (x), treating (y) as (y(x)); consequently, the chain rule gives
  

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    d(y^n)/dx = ny^(n-1) dy/dx.

• General formula: If (F(x,y)=0) and (F_y\ne0), then
  

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    dy/dx = -F_x/F_y.

  
  Here, (F_x) and (F_y) are partial derivatives of (F) with respect to (x) and (y).

• Worked example: For (x^2+y^2=25),
  

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    2x + 2y(dy/dx) = 0
    dy/dx = -x/y.

  
  Thus, the tangent slope depends on both coordinates of the point.

V. Logarithmic Method

A. Logarithmic differentiation

Logarithmic differentiation simplifies products, quotients, variable powers, and functions containing many multiplicative factors.

• Method: Take natural logarithms, use logarithm laws, differentiate implicitly, and multiply by (y) to recover (dy/dx).

• Variable power: For positive (u(x)) and differentiable (v(x)), if (y=u^v), then
  

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    ln y = v ln u
    y'/y = v'ln u + v(u'/u)
    y' = u^v[v'ln u + vu'/u].

• Worked example: For (y=x^x), where (x>0),
  

  TEXT

    ln y = x ln x
    y'/y = ln x + 1
    y' = x^x(ln x+1).

VI. Successive Differentiation

A. Higher order derivatives of simple functions

Higher-order derivatives describe repeated rates of change and are denoted by (f''(x)), (f^{(n)}(x)), or (d^ny/dx^n).

• Standard patterns:
  

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    d^n(x^m)/dx^n = m!x^(m-n)/(m-n)!     for n≤m
    d^n(e^(ax))/dx^n = a^n e^(ax)
    d^n(sin ax)/dx^n = a^n sin(ax+nπ/2)

  
  Here, (m,n) are non-negative integers and (a) is constant.

• Polynomial termination: If (n>m), then the (n)th derivative of (x^m) is zero.

• Engineering interpretation: For position (s(t)), (s'(t)), (s''(t)), and (s'''(t)) represent velocity, acceleration, and jerk respectively.

• Worked example: If (y=e^{2x}), then
  

  TEXT

    y''' = 2^3e^(2x) = 8e^(2x).

VII. Rolle's Theorem

A. Rolle's theorem

Rolle’s theorem guarantees at least one interior point with a horizontal tangent when the endpoint values of a sufficiently regular function are equal.

• Formal statement: If (f) is continuous on ([a,b]), differentiable on ((a,b)), and (f(a)=f(b)), then at least one (c\in(a,b)) satisfies
  

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    f'(c)=0.

  
  Here, (a,b) are endpoints and (c) is an interior point.

• Geometrical meaning: A smooth curve joining two points at the same height must have at least one tangent parallel to the (x)-axis.

• Worked example: For (f(x)=x^2-4x+3) on ([1,3]), (f(1)=f(3)=0), while
  

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    f'(x)=2x-4=0  ⇒  c=2.

• Limitation: The theorem guarantees existence, not uniqueness; failure of any hypothesis removes the guarantee.

VIII. Mean Value Results

A. Mean value theorems

Mean value theorems connect the average change over an interval with an instantaneous change at an interior point.

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

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    f'(c) = {f(b)-f(a)}/(b-a).

  
  The tangent at (c) is parallel to the secant joining the endpoints.

• Cauchy’s mean value theorem: If (f) and (g) are continuous on ([a,b]), differentiable on ((a,b)), and (g'(c)\ne0), then some (c) satisfies
  

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

• Relationship: Rolle’s theorem is the equal-endpoint case of Lagrange’s theorem; Lagrange’s theorem follows from Cauchy’s theorem by taking (g(x)=x).

• Worked example: For (f(x)=x^2) on ([1,3]), the average slope is (4); since (f'(c)=2c), the required point is (c=2).

IX. Series Expansion Theorems

A. Taylor's theorems and Maclaurin theorems

Taylor’s theorem represents a sufficiently differentiable function near a chosen point, while Maclaurin’s theorem is its special case centered at zero.

1. Taylor expansion: About (x=a),
   

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      f(x) = f(a) + (x-a)f'(a) + (x-a)^2f''(a)/2! + ... + R_n.

   
   Here, (R_n) is the remainder after degree (n). In Lagrange form,
   

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      R_n = f^((n+1))(ξ)(x-a)^(n+1)/(n+1)!,

   
   where (\xi) lies between (a) and (x).

2. Maclaurin expansion: Setting (a=0) gives
   

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      f(x) = f(0) + xf'(0) + x^2f''(0)/2! + ...

• Standard series:
  

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    e^x = 1+x+x^2/2!+x^3/3!+...
    sin x = x-x^3/3!+x^5/5!-...
    cos x = 1-x^2/2!+x^4/4!-...
    ln(1+x) = x-x^2/2+x^3/3-...   for |x|<1

• Application: Truncated series provide local polynomial approximations; for small (x), (\sin x\approx x) and (e^x\approx1+x).

X. Undefined Limit Patterns

A. Indeterminate forms

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

• Principal forms:
  

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    0/0,  ∞/∞,  0·∞,  ∞-∞,  0^0,  1^∞,  ∞^0.

• Why indeterminate: The form (0/0) can produce different results; as (x\to0), (x/x\to1), (x^2/x\to0), and (x/x^2) is unbounded.

• Transformations:

  • Convert (0\cdot\infty) into a quotient.
  • Combine terms in (\infty-\infty) using a common denominator or rationalization.
  • For powers, set (y=f(x)^{g(x)}), then examine (\ln y=g(x)\ln f(x)).

• Distinction: Expressions such as (1/0) are undefined or divergent patterns, not automatically indeterminate forms.

XI. Evaluation of Indeterminate Limits

A. L'Hospital's rule

L’Hospital’s rule evaluates certain quotient limits by replacing functions with their derivatives when its hypotheses are satisfied.

• Rule: If (f) and (g) are differentiable near (a), (g'(x)\ne0), and (f/g) has form (0/0) or (\infty/\infty), then
  

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

  
  provided the limit on the right exists or is infinite.

• Repeated use: The rule may be applied again if the differentiated quotient remains (0/0) or (\infty/\infty).

• Worked example:
  

  TEXT

    lim[x→0] (e^x-1-x)/x^2
    = lim[x→0] (e^x-1)/(2x)
    = lim[x→0] e^x/2
    = 1/2.

• Restriction: The derivative of a quotient is not being calculated; numerator and denominator are differentiated separately only within the limit theorem.

XII. Optimization

A. Maxima and minima

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

• Critical points: Interior candidates occur where (f'(c)=0) or (f'(c)) does not exist, provided (f(c)) exists.

• First derivative test:

  • If (f') changes from positive to negative at (c), (f(c)) is a local maximum.
  • If (f') changes from negative to positive, (f(c)) is a local minimum.
  • No sign change means no extremum is established.

• Second derivative test: If (f'(c)=0), then
  

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    f''(c)>0  ⇒ local minimum
    f''(c)<0  ⇒ local maximum.

  
  If (f''(c)=0), the test is inconclusive.

• Absolute extrema: On a closed interval ([a,b]), compare function values at critical points and endpoints.

• Worked example: For (f(x)=x^2-6x+5),
  

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    f'(x)=2x-6=0  ⇒  x=3
    f''(x)=2>0    ⇒  minimum
    f(3)=-4.

  
  Thus, the parabola has an absolute minimum value (-4) at (x=3), with no finite maximum on the real line.