Calculus Essentials: Derivatives and Their Applications

Fundamental Concepts of Differentiation

Relationship Between Differentiability and Continuity

If a function f is differentiable at a point x₀, then f is continuous at x₀. However, the converse is not always true; not all continuous functions are differentiable.

Definition of the Derivative at a Point

The derivative of a function f at a point x₀, denoted f'(x₀), is defined as:

f'(x₀) = lim (∆x→0) (f(x₀ + ∆x) - f(x₀))/∆x

Differential of a Function

For a function y = f(x), its differential dy is given by:

dy = y' * dx = f'(x)dx

Invariance of the Differential

The differential df(x) = f'(x)dx holds true whether x is an independent variable or a function of another variable.

Interpretation of Derivatives and Differentials

The derivative represents the instantaneous rate of change. For example, velocity v(t) is the derivative of displacement d(t) with respect to time:

v(t) = d'(t) = lim (∆t→0) ∆d/∆t

Rules of Differentiation

• Sum Rule: d(u + v) = du + dv
• Product Rule: d(uv) = vdu + udv
• Quotient Rule: d(u/v) = (vdu - udv)/v²

Chain Rule for Composite Functions

The derivative of a composite function f(g(x)) is given by:

(f(g(x)))' = f'(g(x)) * g'(x)

Derivative of an Inverse Function

If f is a differentiable function with a differentiable inverse f⁻¹, then the derivative of the inverse function is:

(f⁻¹(x))' = 1 / f'(f⁻¹(x))

Higher-Order Derivatives

The second derivative of f(x) is f''(x) = (f'(x))'. The n-th derivative of f(x) is f^(n)(x) = (f^(n-1)(x))'.

Example: Derivative of Sine Function from Definition

To find the derivative of f(x) = sin(x) using the definition:

f'(x) = lim (h→0) (sin(x+h) - sin(x))/h

= lim (h→0) (sin(x)cos(h) + cos(x)sin(h) - sin(x))/h

= lim (h→0) (sin(x)(cos(h)-1) + cos(x)sin(h))/h

= lim (h→0) sin(x)(cos(h)-1)/h + lim (h→0) cos(x)sin(h)/h

Since lim (h→0) (cos(h)-1)/h = 0 and lim (h→0) sin(h)/h = 1:

= sin(x) * 0 + cos(x) * 1 = cos(x)

Key Theorems in Differential Calculus

Fermat's Theorem on Local Extrema

If c is an interior point of [a,b], f(x) has a local minimum or maximum at c, and f'(c) exists, then f'(c) = 0.

Rolle's Theorem

If f(x) is continuous on the closed interval [a,b], differentiable on the open interval (a,b), and f(a) = f(b), then there exists at least one point c in (a,b) such that f'(c) = 0.

Lagrange's Mean Value Theorem

If f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists at least one point c in (a,b) such that:

f'(c) = (f(b) - f(a))/(b - a)

Cauchy's Mean Value Theorem

If f(x) and g(x) are continuous on the closed interval [a,b], differentiable on the open interval (a,b), and g'(x) ≠ 0 for all x in (a,b), then there exists at least one point c in (a,b) such that:

f'(c)/g'(c) = (f(b) - f(a))/(g(b) - g(a))

L'Hôpital's Rule

If f(x) and g(x) are differentiable in an open interval containing a (except possibly at a), g'(x) ≠ 0 in that interval, and lim (x→a) f(x) = lim (x→a) g(x) = 0 or lim (x→a) f(x) = lim (x→a) g(x) = ±∞, then:

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

Taylor's Theorem

If f is (n+1) times differentiable in a neighborhood U(a) of a, then for any x in U(a), there exists a c between a and x such that:

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

where R_n(x) = (f^(n+1)(c)/(n+1)!)(x-a)^(n+1) (Lagrange form of the remainder).

Constant Function Theorem

If f is differentiable on (a,b) and f'(x) = 0 for all x ∈ (a,b), then f(x) is a constant function on (a,b).

Applications of Derivatives

Monotonicity of Functions

• A function f(x) is called increasing on a set E ⊆ R if for all x₁ < x₂ in E, f(x₁) ≤ f(x₂).
• A function f(x) is called decreasing on a set E ⊆ R if for all x₁ < x₂ in E, f(x₁) ≥ f(x₂).
• If f'(x) > 0 on an interval, f(x) is strictly increasing on that interval.
• If f'(x) < 0 on an interval, f(x) is strictly decreasing on that interval.

Existence of Extrema: Necessary Condition

If x₀ is a local extremum point (maximum or minimum) and f'(x₀) exists, then f'(x₀) = 0. (This is a direct consequence of Fermat's Theorem).

Second Derivative Test for Local Maxima and Minima

If f'(x₀) = 0:

• If f''(x₀) < 0, then x₀ is a local maximum point.
• If f''(x₀) > 0, then x₀ is a local minimum point.

Conditions for Concavity and Convexity

If f is twice differentiable on (a,b):

• If f''(x) ≤ 0 for all x ∈ (a,b), then f(x) is concave down (or convex) on (a,b).
• If f''(x) ≥ 0 for all x ∈ (a,b), then f(x) is concave up (or concave) on (a,b).

Inflection Points

An inflection point is a point where the concavity of the graph changes (i.e., f''(x) changes sign from positive to negative or vice versa).

Asymptotes: Oblique/Slant Asymptotes

A line y = kx + b is an oblique asymptote to the graph of f(x) as x → ±∞ if:

1. lim (x→±∞) f(x)/x = k (where k is a finite non-zero number)
2. lim (x→±∞) (f(x) - kx) = b (where b is a finite number)

Linear Approximation of Function Values

For small ∆x, the value of f(x + ∆x) can be approximated by:

f(x + ∆x) ≈ f(x) + f'(x)∆x

Equation of the Tangent Line

The equation of the tangent line to the graph of f(x) at the point (a, f(a)) is:

y = f'(a)(x - a) + f(a)

Equation of the Normal Line

The equation of the normal line to the graph of f(x) at the point (x₀, f(x₀)) is:

y = f(x₀) - (1/f'(x₀))(x - x₀), provided f'(x₀) ≠ 0.