Unit 2: Successive Differentiation and Indeterminate Forms

Significance of Derivative and its Sign

The first derivative, f'(x), tells us critical information about the behavior of the original function f(x).

Geometrical Interpretation

As seen in Unit 1, f'(a) is the slope of the tangent line to the curve y = f(x) at the point x = a.

Sign of the First Derivative

The sign of f'(x) determines if the function f(x) is increasing or decreasing on an interval.

• If f'(x) > 0 on an interval (a, b), then f(x) is strictly increasing on (a, b).
• If f'(x) < 0 on an interval (a, b), then f(x) is strictly decreasing on (a, b).
• If f'(x) = 0 on an interval (a, b), then f(x) is constant on (a, b).

Points where f'(x) = 0 or f'(x) is undefined are called critical points. These are potential locations for local maxima or minima.

Derivative as a Rate Measurer

The derivative dy/dx represents the instantaneous rate of change of y with respect to x.

Related Problems (Related Rates)

This is a common application. If two or more quantities are related by an equation and are all changing over time (t), we can find the relationship between their rates of change by differentiating the equation with respect to time.

Problem-Solving Strategy:

1. Identify all given quantities and the quantities to be found. (e.g., given dV/dt, find dr/dt)
2. Write an equation that relates the variables (e.g., V = (4/3)πr³ for a sphere).
3. Differentiate both sides of the equation with respect to time (t), using the Chain Rule. d/dt (V) = d/dt ((4/3)πr³) => dV/dt = 4πr² * (dr/dt)
4. Substitute the known values (for variables and their rates) and solve for the unknown rate.

Successive Differentiation (n-th Derivatives)

Successive differentiation means differentiating a function repeatedly.

• First Derivative: y' or f'(x) or dy/dx
• Second Derivative: y'' or f''(x) or d²y/dx²
• Third Derivative: y''' or f'''(x) or d³y/dx³
• n-th Derivative: yⁿ or fⁿ(x) or dⁿy/dxⁿ

The second derivative, f''(x), has its own significance:

• If f''(x) > 0 on an interval, the graph is concave up (like a cup).
• If f''(x) < 0 on an interval, the graph is concave down (like a frown).

n-th Derivatives of Special Functions

Finding a general formula for the n-th derivative is a key skill. This is usually done by calculating the first few derivatives (f', f'', f''') and looking for a pattern.

n-th Derivatives of Rational Algebraic Functions

To find the n-th derivative of a rational function (a polynomial divided by another polynomial), the key is to first decompose it using partial fractions.

Steps:

1. Check if the fraction is proper (degree of numerator < degree of denominator). If not, perform long division first.
2. Factor the denominator completely.
3. Decompose the rational function into a sum of simpler fractions (e.g., A/(x-a) or (Bx+C)/(x²+b)).
4. Find the n-th derivative of each simple fraction separately, using the standard formulas (like the one for 1/(ax+b)).
5. Add the results.

Example: Find the n-th derivative of y = 1 / (x² - a²).

1. Partial Fractions: 1 / (x² - a²) = 1 / [(x-a)(x+a)] = A/(x-a) + B/(x+a)

   Solving this gives A = 1/(2a) and B = -1/(2a).

   y = [1/(2a)] * [1/(x-a)] - [1/(2a)] * [1/(x+a)]
2. Differentiate:

   We use the formula: dⁿ/dxⁿ [1/(ax+b)] = (-1)ⁿ * n! * aⁿ * (ax+b)⁻(ⁿ⁺¹)

   yⁿ = [1/(2a)] * [(-1)ⁿ n! (1)ⁿ (x-a)⁻(ⁿ⁺¹)] - [1/(2a)] * [(-1)ⁿ n! (1)ⁿ (x+a)⁻(ⁿ⁺¹)]
3. Simplify: yⁿ = [(-1)ⁿ n! / (2a)] * [ 1/(x-a)ⁿ⁺¹ - 1/(x+a)ⁿ⁺¹ ]

Leibnitz's Theorem

This theorem provides a formula for the n-th derivative of a product of two functions.

If y = u * v, where u and v are functions of x, then the n-th derivative of y is: (uv)ⁿ = ⁿC₀ uⁿv + ⁿC₁ uⁿ⁻¹v¹ + ⁿC₂ uⁿ⁻²v² + ... + ⁿCᵣ uⁿ⁻ʳvʳ + ... + ⁿCₙ uvⁿ Where:

• uⁿ is the n-th derivative of u.
• vʳ is the r-th derivative of v.
• ⁿCᵣ = n! / [r! * (n-r)!] is the binomial coefficient.

Indeterminate Forms

When evaluating a limit, we sometimes get an undefined expression. These are called indeterminate forms. L'Hospital's Rule is a tool to solve limits that result in these forms.

The main indeterminate forms are:

• 0 / 0
• ∞ / ∞

Other forms can be converted into these two:

• 0 * ∞ (Convert to 0/0 or ∞/∞. e.g., f*g = f / (1/g))
• ∞ - ∞ (Combine terms, e.g., by finding a common denominator)
• 1^∞
• 0⁰
• ∞⁰

For the exponential forms (1^∞, 0⁰, ∞⁰), let the limit be L. Take the natural logarithm (ln L) on both sides. This will turn the limit into a 0 * ∞ form, which can then be converted to 0/0 or ∞/∞.

L'Hospital's Theorem (or Rule)

L'Hospital's Rule is a method for evaluating limits that result in the indeterminate forms 0/0 or ∞/∞.

Suppose lim (x → a) f(x) = 0 and lim (x → a) g(x) = 0.
OR
Suppose lim (x → a) f(x) = ±∞ and lim (x → a) g(x) = ±∞.

Then: lim (x → a) [f(x) / g(x)] = lim (x → a) [f'(x) / g'(x)] ...provided the limit on the right side exists (or is ±∞).

Important Notes:

• The rule applies for x → a, x → a⁺, x → a⁻, x → ∞, or x → -∞.
• Common Mistake: Do NOT use the quotient rule. You differentiate the numerator and denominator separately.
• If after applying the rule once, you still get 0/0 or ∞/∞, you can apply the rule again (and again) until the limit can be evaluated.
• Check the condition! The rule ONLY works for 0/0 or ∞/∞. If you apply it to a limit that is, for example, 1/2, you will get the wrong answer.

Example (Exponential Form 1^∞)

Evaluate L = lim (x → 0) (1 + x)¹/ˣ

1. This is of the form 1^∞.
2. Take the natural log: ln L = ln [ lim (x → 0) (1 + x)¹/ˣ ] ln L = lim (x → 0) [ ln( (1 + x)¹/ˣ ) ] ln L = lim (x → 0) [ (1/x) * ln(1 + x) ] ln L = lim (x → 0) [ ln(1 + x) / x ]
3. Now, this is a 0/0 form. Apply L'Hospital's Rule: ln L = lim (x → 0) [ d/dx(ln(1+x)) / d/dx(x) ] ln L = lim (x → 0) [ (1 / (1+x)) / 1 ] ln L = lim (x → 0) 1 / (1 + x) = 1 / (1 + 0) = 1
4. We found ln L, not L. ln L = 1 => L = e¹ = e