Unit 3: Differentiability of function

Differentiability studies the rate at which a function changes and the conditions under which that rate is well defined. The derivative of f at x = a is the limit of the average rate of change over a shrinking interval, and it exists only when the function is smooth enough at that point.

Governing definition: the derivative is

f'(a) = lim(h→0) [f(a+h) − f(a)] / h

provided the limit exists and is finite.

• Domain of the derivative: f'(x) is itself a function, defined only where the limit exists.
• Geometric meaning: f'(a) is the slope of the tangent to y = f(x) at x = a.
• Continuity is necessary: if f is differentiable at a, it is continuous at a; the converse fails (e.g. |x| at 0).
• Notation: f'(x), dy/dx, and D f all denote the same object; the operator d/dx acts on a function.
• Standard forms carried throughout: d/dx(x^n) = n·x^(n−1), d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(constant) = 0.

II. Foundations of the Derivative

Definition, existence, and the arithmetic rules.

A. Derivatives by first principle

This is the direct evaluation of the defining limit, without shortcut rules.

• Method: form the difference quotient [f(x+h) − f(x)] / h, simplify to cancel the h in the denominator, then let h → 0.
• Purpose: it proves the standard formulae and handles functions with no ready rule.
• Worked example — f(x) = x²:
  
  TEXT

  f'(x) = lim(h→0) [(x+h)² − x²]/h
        = lim(h→0) [2xh + h²]/h
        = lim(h→0) (2x + h) = 2x

• Trigonometric case: for sin x, expanding sin(x+h) and using lim(h→0)(sin h)/h = 1 and lim(h→0)(1−cos h)/h = 0 gives cos x.

B. Algebra of derivative of functions

Derivatives distribute over the basic arithmetic operations by fixed rules.

• Sum/difference: (u ± v)' = u' ± v'.
• Constant multiple: (k·u)' = k·u' for constant k.
• Product rule: (u·v)' = u'·v + u·v'.
• Quotient rule: (u/v)' = (u'·v − u·v') / v², valid where v ≠ 0.
• Anchor: for y = x²·sin x, y' = 2x·sin x + x²·cos x.

C. Differentiability

A function is differentiable at a point when its left- and right-hand derivatives agree.

• Left derivative: lim(h→0⁻) [f(a+h) − f(a)]/h.
• Right derivative: lim(h→0⁺) [f(a+h) − f(a)]/h.
• Criterion: differentiable at a ⇔ both one-sided derivatives exist and are equal.
• Failure modes: a corner (|x| at 0, slopes −1 and +1), a cusp, a vertical tangent (x^(1/3) at 0), or a discontinuity.
• On an interval: f is differentiable on (a,b) if f'(x) exists for every point of the interval.

III. Differentiating Combinations of Functions

Composition and relations that are not solved for y.

A. Derivatives of composite functions

A composite f(g(x)) is differentiated by peeling the outer function first while holding the inner intact.

• Structure: identify outer f and inner g so that y = f(u), u = g(x).
• Principle: differentiate the outer function at the inner value, then multiply by the derivative of the inner.
• Anchor: for y = (3x + 1)⁵, treat u = 3x + 1, giving y' = 5(3x+1)⁴·3.

B. Chain rule

The chain rule is the formal statement that governs composite differentiation.

• Formula:
  
  TEXT

  dy/dx = (dy/du) · (du/dx)

  
  where y = f(u) and u = g(x).
• Extension: for triple composition y = f(g(h(x))),
  
  TEXT

  dy/dx = f'(g(h(x))) · g'(h(x)) · h'(x)

• Worked example — y = sin(x²): with u = x², dy/du = cos u, du/dx = 2x, so y' = 2x·cos(x²).

C. Derivatives of implicit functions

When y is defined implicitly by an equation F(x, y) = 0, differentiate every term with respect to x, treating y as a function of x.

• Key step: each y-term picks up a factor dy/dx by the chain rule.
• Then: collect all dy/dx terms on one side and solve algebraically.
• Worked example — x² + y² = 25:
  
  TEXT

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

• Use: essential for curves like x²+y²=r² or xy = c that cannot be written as y = f(x) globally.

IV. Derivatives of Special Functions

Inverse trigonometric, exponential, logarithmic, and the logarithmic technique.

A. Derivatives of inverse trigonometric functions

These follow from the chain rule applied to the defining relation (e.g. y = sin⁻¹x ⇔ sin y = x).

• Standard results:
  
  TEXT

  d/dx(sin⁻¹x) =  1/√(1−x²)      |x| < 1
  d/dx(cos⁻¹x) = −1/√(1−x²)      |x| < 1
  d/dx(tan⁻¹x) =  1/(1+x²)
  d/dx(cot⁻¹x) = −1/(1+x²)
  d/dx(sec⁻¹x) =  1/(|x|√(x²−1))  |x| > 1
  d/dx(csc⁻¹x) = −1/(|x|√(x²−1))  |x| > 1

• Derivation sketch: differentiate sin y = x implicitly: cos y·(dy/dx) = 1, so dy/dx = 1/cos y = 1/√(1−x²).
• Anchor: paired functions differ only in sign; sin⁻¹ and cos⁻¹ derivatives sum to zero.

B. Derivatives of exponential and logarithmic functions

The exponential base e is the fixed point of differentiation, and the logarithm is its inverse.

• Core results:
  
  TEXT

  d/dx(e^x) = e^x
  d/dx(a^x) = a^x·ln a           a > 0
  d/dx(ln x) = 1/x               x > 0
  d/dx(log_a x) = 1/(x·ln a)

• With chain rule: d/dx(e^(g(x))) = e^(g(x))·g'(x); e.g. d/dx(e^(2x)) = 2e^(2x).
• Anchor: a^x = e^(x ln a), so its derivative inherits the factor ln a.

C. Logarithmic differentiation

This technique takes the natural log of both sides before differentiating, simplifying products, quotients, and variable exponents.

• When to use: functions of the form y = [f(x)]^(g(x)), or long products/quotients.
• Procedure: take ln of both sides, use log laws to split, differentiate implicitly, then multiply through by y.
• Worked example — y = x^x:
  
  TEXT

  ln y = x·ln x
  (1/y)·(dy/dx) = ln x + 1
  dy/dx = x^x (ln x + 1)

• Anchor: neither the power rule nor the exponential rule alone handles a variable base with a variable exponent.

V. Parametric and Higher-Order Derivatives

Curves given through a parameter, and repeated differentiation.

A. Derivatives of functions in parametric forms

When x and y are each given as functions of a parameter t, the derivative is a ratio of parameter derivatives.

• Formula:
  
  TEXT

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

• Reason: by the chain rule, dy/dt = (dy/dx)·(dx/dt), then rearrange.
• Worked example — x = a cos t, y = a sin t:
  
  TEXT

  dx/dt = −a sin t,  dy/dt = a cos t
  dy/dx = (a cos t)/(−a sin t) = −cot t

• Anchor: the result is expressed in terms of t, not x.

B. Second order derivatives

The second derivative is the derivative of the first derivative and measures how the slope itself changes.

• Notation: f''(x), d²y/dx², or D²y.
• Meaning: it gives concavity — f'' > 0 means the curve bends upward (convex), f'' < 0 means it bends downward.
• Direct case: for y = x⁴, y' = 4x³, y'' = 12x².
• Parametric case (needs care): differentiate dy/dx (a function of t) with respect to t, then divide again by dx/dt:
  
  TEXT

  d²y/dx² = d/dt(dy/dx) · (1/(dx/dt))

  
  It is not (d²y/dt²)/(d²x/dt²).
• Implicit case: differentiate dy/dx again with respect to x, substituting the known dy/dx where it reappears.
• Anchor: for x²+y²=25, from dy/dx = −x/y,
  
  TEXT

  d²y/dx² = −(y − x·(dy/dx))/y² = −(y + x²/y)/y² = −25/y³

The derivative, built from the first-principle limit and extended by the algebra, chain, implicit, and logarithmic techniques, reduces almost every differentiation problem to a small set of standard forms combined by fixed rules; the second derivative then reapplies the same machinery to describe curvature.