Differential Calculus I: Successive Derivatives, Leibniz & Curves

Unit 2 — Differential Calculus I (8 Hours)

This unit introduces advanced differentiation techniques and their applications to functions and curves, forming the mathematical backbone of engineering analysis, optimization, and modeling.

1. What Is Successive Differentiation?

Successive differentiation is the process of differentiating a function repeatedly to obtain higher-order derivatives.

If $y = f(x)$ , then:

First derivative: $\displaystyle \frac{dy}{dx} = f'(x) = y_1$
Second derivative: $\displaystyle \frac{d^2y}{dx^2} = f''(x) = y_2$
$n$ -th derivative: $\displaystyle \frac{d^ny}{dx^n} = f^{(n)}(x) = y_n$

Why It Matters

Acceleration is the second derivative of displacement
Bending moments in beams involve fourth derivatives of deflection
Taylor series expansions require higher-order derivatives
Stability analysis in control systems uses higher derivatives

2. Standard Formulas for Successive Differentiation

2.1 Power Function

If $y = x^m$ :

y_n = m(m-1)(m-2)\cdots(m-n+1) \, x^{m-n} = \frac{m!}{(m-n)!} x^{m-n}

Special case when $m = n$ :

\frac{d^n}{dx^n}(x^n) = n!

If $m$ is a positive integer and $n > m$ :

y_n = 0

2.2 Exponential Function

If $y = e^{ax}$ :

y_n = a^n e^{ax}

If $y = a^x$ :

y_n = a^x (\ln a)^n

2.3 Logarithmic Function

If $y = \ln(x)$ :

y_n = \frac{(-1)^{n-1} (n-1)!}{x^n}

If $y = \ln(ax + b)$ :

y_n = \frac{(-1)^{n-1} a^n (n-1)!}{(ax + b)^n}

2.4 Trigonometric Functions

If $y = \sin(ax + b)$ :

y_n = a^n \sin\left(ax + b + \frac{n\pi}{2}\right)

If $y = \cos(ax + b)$ :

y_n = a^n \cos\left(ax + b + \frac{n\pi}{2}\right)

2.5 Product with Exponential

If $y = e^{ax} \sin(bx + c)$ :

y_n = r^n e^{ax} \sin(bx + c + n\theta)

where $r = \sqrt{a^2 + b^2}$ and .

If $y = e^{ax} \cos(bx + c)$ :

y_n = r^n e^{ax} \cos(bx + c + n\theta)

3. Leibniz Theorem

3.1 Statement

If $u$ and $v$ are functions of $x$ with derivatives of all orders, then the $n$ -th derivative of their product is:

\frac{d^n}{dx^n}(uv) = \sum_{r=0}^{n} \binom{n}{r} u_{n-r} \, v_r

Expanded form:

(uv)_n = u_n v + \binom{n}{1} u_{n-1} v_1 + \binom{n}{2} u_{n-2} v_2 + \cdots + \binom{n}{r} u_{n-r} v_r + \cdots + u v_n

where $u_k = \frac{d^k u}{dx^k}$ and .

3.2 Key Insight

Choose $v$ as the function whose derivatives eventually vanish (e.g., polynomials) to limit the number of terms.

3.3 Application Example

To find the $n$ -th derivative of $x^2 e^{ax}$ :

Let $u = e^{ax}$ (derivatives never vanish)
Let $v = x^2$ (third and higher derivatives are zero)

Only three terms survive:

\frac{d^n}{dx^n}(x^2 e^{ax}) = a^n e^{ax} x^2 + n \cdot a^{n-1} e^{ax} \cdot 2x + \frac{n(n-1)}{2} \cdot a^{n-2} e^{ax} \cdot 2

4. Curve Tracing

4.1 Important Features of Curves

When sketching $y = f(x)$ or $f(x, y) = 0$ , analyze:

Feature	What to Check
Domain & Range	Where is the function defined?
Intercepts	Where does it cross the axes?
Symmetry	Even, odd, or periodic?
Asymptotes	Where does the curve approach but never reach?
Critical Points	Where is $\frac{dy}{dx} = 0$ or undefined?

4.2 Symmetry

Condition	Symmetry
$f(-x, y) = f(x, y)$	About $y$ -axis
$f (x, - y) = f$

4.3 Intercepts

$x$ -intercepts: Set $y = 0$ , solve for $x$
$y$ -intercepts: Set $x = 0$ , solve for

4.4 Asymptotes

An asymptote is a straight line that a curve approaches as $x \to \infty$ or $y \to \infty$ .

Vertical Asymptotes: Occur where $y \to \infty$ . Find values of $x$ that make the denominator zero (after canceling common factors).

Horizontal Asymptotes:

y = L \quad \text{where} \quad L = \lim_{x \to \infty} f(x)

Oblique (Slant) Asymptotes: If degree of numerator = degree of denominator + 1, perform polynomial long division:

y = mx + c \quad \text{where} \quad m = \lim_{x \to \infty} \frac{f(x)}{x}, \quad c = \lim_{x \to \infty} [f(x) - mx]

4.5 Maximum and Minimum Points

First Derivative Test:

If $\frac{dy}{dx}$ changes from $+$ to $-$ at $x = c$ , then is a

Second Derivative Test: At a critical point $x = c$ where $f'(c) = 0$ :

If $f''(c) < 0$ : Local maximum
If $f''(c) > 0$ :

4.6 Points of Inflection

A point where the concavity changes:

\frac{d^2y}{dx^2} = 0 \quad \text{and changes sign}

5. Partial Derivatives

5.1 Functions of Several Variables

A function $f(x, y)$ depends on multiple independent variables.

5.2 First-Order Partial Derivatives

Partial derivative with respect to $x$ (treat $y$ as constant):

\frac{\partial f}{\partial x} = f_x = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}

Partial derivative with respect to $y$ (treat $x$ as constant):

\frac{\partial f}{\partial y} = f_y = \lim_{k \to 0} \frac{f(x, y+k) - f(x, y)}{k}

5.3 Higher-Order Partial Derivatives

\frac{\partial^2 f}{\partial x^2} = f_{xx}, \quad \frac{\partial^2 f}{\partial y^2} = f_{yy}

5.4 Mixed Partial Derivatives

\frac{\partial^2 f}{\partial y \, \partial x} = f_{xy} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)

\frac{\partial^2 f}{\partial x \, \partial y} = f_{yx} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)

Schwarz's Theorem (Clairaut's Theorem): If $f_{xy}$ and $f_{yx}$ are continuous at a point, then:

\frac{\partial^2 f}{\partial y \, \partial x} = \frac{\partial^2 f}{\partial x \, \partial y}

5.5 Notation Summary

Derivative	Symbol
$\frac{\partial f}{\partial x}$	$f_x$ ,

6. Euler's Theorem for Homogeneous Functions

6.1 Homogeneous Functions

A function $f(x, y)$ is homogeneous of degree $n$ if:

f(tx, ty) = t^n f(x, y)

for all $t > 0$ .

Test: In a homogeneous function, each term has the same total degree.

Example: $f(x, y) = x^3 + 3x^2y + y^3$ is homogeneous of degree 3.

6.2 Euler's Theorem

If $f(x, y)$ is a homogeneous function of degree $n$ with continuous partial derivatives, then:

x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y)

In operator notation:

x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y} = n

when applied to a homogeneous function of degree $n$ .

6.3 Extension to Second Derivatives

For a homogeneous function of degree $n$ :

x^2 \frac{\partial^2 f}{\partial x^2} + 2xy \frac{\partial^2 f}{\partial x \, \partial y} + y^2 \frac{\partial^2 f}{\partial y^2} = n(n-1) f(x, y)

Or using operators:

\left(x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}\right)^2 f = n(n-1) f

6.4 General Form for $m$ -Variables

If $f(x_1, x_2, \ldots, x_m)$ is homogeneous of degree :

\sum_{i=1}^{m} x_i \frac{\partial f}{\partial x_i} = n f

7. Total Derivative

7.1 Total Differential

If $z = f(x, y)$ where $x$ and $y$ are independent variables, the total differential is:

dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy

7.2 Total Derivative (Chain Rule)

If $z = f(x, y)$ where $x = x(t)$ and $y = y(t)$ :

\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}

If $z = f(x, y)$ where $y = g(x)$ :

\frac{dz}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx}

7.3 Implicit Differentiation

If $f(x, y) = c$ (constant), then:

\frac{dy}{dx} = -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} = -\frac{f_x}{f_y}

provided $f_y \neq 0$ .

7.4 Second Total Derivative

\frac{d^2z}{dt^2} = \frac{\partial^2 f}{\partial x^2}\left(\frac{dx}{dt}\right)^2 + 2\frac{\partial^2 f}{\partial x \, \partial y}\frac{dx}{dt}\frac{dy}{dt} + \frac{\partial^2 f}{\partial y^2}\left(\frac{dy}{dt}\right)^2 + \frac{\partial f}{\partial x}\frac{d^2x}{dt^2} + \frac{\partial f}{\partial y}\frac{d^2y}{dt^2}

8. Change of Variables

8.1 Transformation of Variables

When changing from $(x, y)$ to new variables $(u, v)$ where $u = u(x, y)$ and $v$ :

\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial x}

\frac{\partial f}{\partial y} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial y}

8.2 Jacobian Determinant

The Jacobian of the transformation is:

J = \frac{\partial(u, v)}{\partial(x, y)} = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}

If $J \neq 0$ , the transformation is locally invertible.

For inverse transformation:

\frac{\partial(x, y)}{\partial(u, v)} = \frac{1}{\frac{\partial(u, v)}{\partial(x, y)}} = \frac{1}{J}

8.3 Polar Coordinates Transformation

A common change of variables: $x = r\cos\theta$ , $y = r\sin\theta$

\frac{\partial f}{\partial r} = \frac{\partial f}{\partial x}\cos\theta + \frac{\partial f}{\partial y}\sin\theta

\frac{\partial f}{\partial \theta} = -\frac{\partial f}{\partial x} r\sin\theta + \frac{\partial f}{\partial y} r\cos\theta

Inverse relations:

\frac{\partial f}{\partial x} = \frac{\partial f}{\partial r}\cos\theta - \frac{1}{r}\frac{\partial f}{\partial \theta}\sin\theta

\frac{\partial f}{\partial y} = \frac{\partial f}{\partial r}\sin\theta + \frac{1}{r}\frac{\partial f}{\partial \theta}\cos\theta

9. Engineering Applications

Domain	Application
Structural Engineering	Beam deflection: $EI \frac{d^4y}{dx^4} = w(x)$

10. Common Mistakes to Avoid

Mistake	Correction
Treating $\frac{dy}{dx}$ as a fraction in partial derivatives	Partial derivatives cannot be casually "canceled"
Assuming $f_{xy} = f_{yx}$ without continuity check

11. Best Practices & Architecture

Always verify continuity before equating mixed partials
Use logarithmic differentiation for products/quotients of many terms
Check homogeneity by substituting $(tx, ty)$ before applying Euler's theorem
Draw sign charts for $f'$ and $f''$ when curve tracing

12. Practice Problems

Problem 1: Find the $n$ -th derivative of $y = \frac{1}{ax + b}$ .

Problem 2: Using Leibniz theorem, find the $n$ -th derivative of $x^3 \sin x$ .

Problem 3: Trace the curve $y = \frac{x^2}{x^2 - 1}$ , identifying all asymptotes, intercepts, and extrema.

Problem 4: If $u = \sin^{-1}\left(\frac{x + y}{\sqrt{x} + \sqrt{y}}\right)$ , prove that .

Problem 5: If $z = f(x, y)$ where $x = e^u + e^{-v}$ and , show that:

\frac{\partial z}{\partial u} - \frac{\partial z}{\partial v} = x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y}

Problem 6: If $V = x^2 y + y^2 z + z^2 x$ , verify that .

13. Mini Project

Title: Optimization of a Cantilever Beam Profile

Objective: Use successive differentiation and partial derivatives to optimize the cross-sectional dimensions of a cantilever beam to minimize weight while satisfying stress constraints.

Given:

Beam length $L$ , load $P$ at free end
Rectangular cross-section: width $b$ , height $h$
Maximum bending stress: $\sigma_{max} = \frac{6PL}{bh^2} \leq \sigma_{allow}$

Tasks:

Express weight $W$ as a function of $b$ and $h$ with the stress constraint
Use partial derivatives to find critical points of the Lagrangian
Verify minimum using second derivative test (Hessian matrix)
Find optimal $b/h$ ratio
Compute successive derivatives of the deflection curve $y(x) = \frac{Px^2}{6EI}(3L - x)$ to verify boundary conditions

Deliverable: A computational notebook (Python/MATLAB) that symbolically derives the optimality conditions, visualizes the feasible region, and validates the solution using curve tracing of constraint boundaries.

Quick Reference Formula Sheet

Concept	Formula
$n$ -th derivative of $x^m$	$\frac{m!}{(m-n)!}x^{m-n}$

End of Unit 2 — Differential Calculus I