Differential Calculus Compact Note

math-1109 MATH-1109 Differential and Integral Calculus

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Comprehensive Note on Differentiation and Calculus Concepts

The notes cover several major areas of calculus, including foundations (Limits, Continuity, Differentiability), advanced differentiation techniques (Successive, Partial), related theorems (Euler’s, Mean Value, Taylor’s), and applications (Tangent/Normal, Maxima/Minima, Asymptotes, Curvature).

I. Foundations: Limit, Continuity, and Differentiability

A. Limit

The concept of a limit involves a limiting value and a functional value.

Definition: A function $f(x)$ is said to have a limit $l$ at $x=a$ if for every $h>0$ (sufficiently small), there exists $\delta >0$ ($\delta$ depending on $h$) such that $|f(x)-l|<h$ whenever $0<|x-a|<\delta$.

Mathematical Notation: ${\lim }_{x\to a}f(x)=l$.

Existence Condition: A limit exists if and only if the Left Hand Limit (LHL) and the Right Hand Limit (RHL) are equal and finite.

Component	Notation
Left Hand Limit (LHL)	${\lim }_{x\to a^{-}}f(x)=l_1$
Right Hand Limit (RHL)	${\lim }_{x\to a^{+}}f(x)=l_2$

B. Continuity

Definition: A function $f(x)$ is continuous at $x=a$ if the limit exists, is finite, and is equal to the functional value.

Mathematical Condition: ${\lim }_{x\to a}f(x)=f(a)$. Condition via LHL/RHL: LHL = RHL = $f(a)$.

C. Differentiation / Derivative / Differentiability

Definition: The process of finding the differential coefficient of $y=f(x)$ with respect to $x$ is called the derivative or differentiation of $f(x)$ with respect to $x$.

Differential Coefficient ($f^{\prime }(x)$ or $dy/dx$): $f^{\prime }(x)=\frac{dy}{dx}={\lim }_{\Delta x\to 0}\frac{\Delta y}{\Delta x}={\lim }_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

Existence Condition (LHD and RHD): The derivative $f^{\prime }(x)$ exists at $x=a$ if the Left Hand Derivative (LHD) equals the Right Hand Derivative (RHD).

Type	Formula
Left Hand Derivative (LHD)	$L{f}^{\prime }(a)={\lim }_{h\to 0}\frac{f(a-h)-f(a)}{-h}$
Right Hand Derivative (RHD)	$R{f}^{\prime }(a)={\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$

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II. Advanced Differentiation Techniques

A. Successive Differentiation

Successive differentiation involves finding higher-order derivatives of a function.

Leibnitz’s Theorem: This theorem provides the formula for finding the $n^{th}$ derivative of the product of two functions, $u$ and $v$ (which are functions of $x$ only).

Formula for $(UV{)}_n$: $(UV{)}_n=U_{n}V+{}^n{C}_1{U}_{n-1}{V}_1+{}^n{C}_2{U}_{n-2}{V}_2+\cdots +{}^n{C}_r{U}_{n-r}{V}_r+\cdots +U{V}_n$ *(Note: The suffix of $U$ and $V$ denotes the order of differentiation with respect to $x$).

B. Partial Differentiation

Definition: The result of differentiating a function $u=f(x,y)$ with respect to $x$, while treating $y$ as a constant, is called partial differentiation of $u$ with respect to $x$.

Notation: Partial derivatives are denoted by symbols such as $\frac{\partial u}{\partial x}$, $u_x$, $\frac{\partial f}{\partial x}$, or $f_x$.

C. Total Differentials

For a function $u=f(x,y)$ where $x$ and $y$ are parametric equations of $t$: $\frac{du}{dt}=\frac{\partial u}{\partial x}\frac{dx}{dt}+\frac{\partial u}{\partial y}\frac{dy}{dt}$

Total Differential ($du$):

• If $u=f(x,y)$: $du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy$.
• If $u=f(x,y,z)$: $du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy+\frac{\partial u}{\partial z}dz$.

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III. Homogeneous Functions and Euler’s Theorem

A. Homogeneous Function

Definition: A function $f(x,y)$ is homogeneous of degree $n$ in $x$ and $y$ if it can be written in the form $x^n\Phi (y/x)$ or $y^n\Psi (x/y)$.

B. Euler’s Theorem

Formula (Two variables): If $f(x,y)$ is a homogeneous function of degree $n$, then: $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf$

Formula (Three variables): If $f(x,y,z)$ is a homogeneous function of degree $n$, then: $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}+z\frac{\partial f}{\partial z}=nf$

For Composite Functions: If $V=f(u)$, where $u$ is a homogeneous function of degree $n$ in $x$ and $y$, then: $x\frac{\partial V}{\partial x}+y\frac{\partial V}{\partial y}=nu\frac{dV}{du}$

Property of Derivatives: If $V$ is a homogeneous function of degree $n$ in $x,y,z$, then its first partial derivatives ($\frac{\partial V}{\partial x},\frac{\partial V}{\partial y},\frac{\partial V}{\partial z}$) are each homogeneous functions of degree $n-1$ in $x,y,z$.

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IV. Applications of Derivatives

A. Maximum and Minimum

Definition of Maximum Value: A function $f(x)$ has a maximum value at $x=a$ if there exists a positive quantity $h$ such that for all $x$ in the interval $a-h\le x\le a+h$ ($x\ne a$), $f(a)>f(x)$.

Definition of Minimum Value: A function $f(x)$ has a minimum value at $x=b$ if there exists a positive quantity $h$ such that for all $x$ in the interval $b-h\le x\le b+h$ ($x\ne b$), $f(x)<f(b)$.

Determination Rules (Second Derivative Test):

1. Maximum: If $f^{\prime \prime }(a_1)<0$.
2. Minimum: If $f^{\prime \prime }(a_1)>0$.
3. Neither Max nor Min (Inflection Point): If $f^{\prime \prime }(a_1)=0$ AND $f^{\prime \prime \prime }(a_1)\ne 0$.

B. Indeterminate Forms and L’Hospital’s Theorem

L’Hospital’s Theorem is used to evaluate limits that result in indeterminate forms, such as $\frac{0}{0}$. The rule states that if ${\lim }_{x\to a}\frac{f(x)}{g(x)}$ results in $\frac{0}{0}$ form, the limit can be found by evaluating ${\lim }_{x\to a}\frac{f^{\prime }(x)}{g^{\prime }(x)}$. This process can be repeated if the derivative still yields an indeterminate form.

The sources also demonstrate methods for handling other indeterminate forms, like $0\times \infty$.

C. Tangent and Normal

Tangent Definition: The tangent is the limiting position of a secant PQ as point Q approaches point P along the curve. Normal Definition: The normal is the straight line drawn perpendicular to the tangent at that point.

Function Type	Equation of Tangent	Equation of Normal
Explicit ($y=f(x)$)	$Y-y=\frac{dy}{dx}(X-x)$	$\frac{dy}{dx}(Y-y)+(X-x)=0$
Implicit ($f(x,y)=0$ at $(x,y)$)	$(X-x)f_x+(Y-y)f_y=0$	$\frac{X-x}{f_x}=\frac{Y-y}{f_y}$

D. Subtangent and Subnormal

The definitions relate to the projections onto the x-axis.

• Subtangent: The projection TM of the tangent PT on the x-axis. $\text{Subtangent}=\frac{Y}{dy/dx}$
• Subnormal: The projection MN of the normal PN on the x-axis. $\text{Subnormal}=Y\frac{dy}{dx}$

E. Angle of Intersection (between $f(x,y)=0$ and $\Phi (x,y)=0$)

The angle of intersection $\alpha$ between two implicit curves is given by: $\tan \alpha =\pm \frac{f_x{\Phi }_y-f_y{\Phi }_x}{f_x{\Phi }_x+f_y{\Phi }_y}$

• Orthogonal Cut ($\alpha =90^{\circ }$): Two curves cut orthogonally if $f_x{\Phi }_x+f_y{\Phi }_y=0$.
• Curves Touch Each Other ($\alpha =0^{\circ }$): Two curves touch each other if $f_x{\Phi }_y-f_y{\Phi }_x=0$.

F. Asymptotes

Definition: A straight line is called an asymptote to a curve if it meets the curve in two coincident points at infinity and is not wholly at infinity itself.

Finding Oblique Asymptotes ($y=mx+c$):

1. Slope ($m$): Found by solving ${\Phi }_n(m)=0$.
2. Intercept ($c$) (for distinct roots $m$): Found using $c=-\frac{{\Phi }_{n-1}(m)}{{\Phi }_n^{\prime }(m)}$.
3. Intercept ($c$) (for repeated roots $m$): Found by solving the quadratic equation: $\frac{c^2}{2!}{\Phi }_n^{\prime \prime }(m)+c{\Phi }_{n-1}^{\prime }(m)+{\Phi }_{n-2}(m)=0$.

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V. Expansion Functions and Theorems

A. Rolle’s Theorem

Conditions:

1. $f(x)$ is continuous in the closed interval $a\le x\le b$ (or $x\in [a,b]$).
2. $f^{\prime }(x)$ exists in the open interval $a<x<b$ (or $x\in (a,b)$).
3. $f(a)=f(b)$.

Conclusion: There exists at least one value of $x$ (say $c$) between $a$ and $b$ such that $f^{\prime }(c)=0$.

B. Mean Value Theorem

Conditions:

1. $f(x)$ is continuous in the closed interval $a\le x\le b$ (or $x\in [a,b]$).
2. $f^{\prime }(x)$ exists in the open interval $a<x<b$ (or $x\in (a,b)$).

Conclusion: There exists at least one value of $x$ (say $c$) between $a$ and $b$ such that $f(b)-f(a)=(b-a)f^{\prime }(c)$.

C. Taylor’s Theorem

Conditions: If $f(x)$ has differential coefficients of the first $(n-1)$ order for every value of $x$ in the closed interval $a\le x\le b$, and the $n^{th}$ derivative exists in the open interval $a<x<b$.

Formula (Taylor Series Expansion): $f(x)=f(a)+(x-a)f^{\prime }(a)+\frac{(x-a{)}^2}{2!}{f}^{\prime \prime }(a)+\frac{(x-a{)}^3}{3!}{f}^{\prime \prime \prime }(a)+\cdots$

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VI. Curvature

A. Definitions

• Angle of Contignance: $\delta \Psi$.
• Arc Length: $\delta s$.
• Average Curvature (Average Bending): The ratio of the corresponding angle of contignance ($\delta \Psi$) to the length of the arc ($\delta s$).
• Curvature ($\kappa$): The limit of the average curvature as the arc length approaches zero. $\kappa ={\lim }_{\delta s\to 0}\frac{\delta \Psi }{\delta s}=\frac{d\Psi }{ds}$.
• Radius of Curvature ($\rho$): The reciprocal of the curvature ($\kappa$). $\rho =\frac{1}{\kappa }=\frac{ds}{d\Psi }$

B. Formulas for Radius of Curvature ($\rho$)

1. Explicit Function $y=f(x)$

$$\rho =\frac{{\left(1+{\left(\frac{dy}{dx}\right)}^2\right)}^{3/2}}{\left|\frac{d^{2}y}{d{x}^2}\right|}$$

2. Implicit Function $F(x,y)=0$

$$\rho =\frac{{\left(F_x^2+F_y^2\right)}^{3/2}}{\left|F_x^2{F}_{yy}-2F_x{F}_y{F}_{xy}+F_y^2{F}_{xx}\right|}$$

3. Parametric Equations $x=\Phi (t),,y=\Psi (t)$

$$\rho =\frac{{\left({\dot{x}}^2+{\dot{y}}^2\right)}^{3/2}}{\left|\dot{x}\ddot{y}-\dot{y}\ddot{x}\right|}$$

4. Polar Function $r=f(\theta )$

$$\rho =\frac{{\left(r^2+{\left(\frac{dr}{d\theta }\right)}^2\right)}^{3/2}}{\left|,r^2+2{\left(\frac{dr}{d\theta }\right)}^2-r\frac{d^{2}r}{d{\theta }^2},\right|}$$

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VII. Tricks and Tips (Not from Sources)

Here are some helpful tips for tackling these mathematical concepts:

1. Identify the Function Type Immediately: When faced with a differentiation problem, quickly determine if it is Explicit ($y$ isolated), Implicit ($x$ and $y$ mixed), or Parametric (both $x$ and $y$ defined by a third variable, $t$ or $\theta$). This dictates which formula for Radius of Curvature, Tangent, or Normal you must use.

2. Logarithmic Differentiation for Exponents: If a function has a variable base raised to a variable power (e.g., $y=(\cos x{)}^{\sin x}$ shown in the source), or if it involves complex products/quotients, always take the natural logarithm ($\ln$ or $\log$) of both sides first. This converts multiplication and exponentiation into simpler addition and multiplication, making differentiation via the chain rule easier.

3. Differentiability Hierarchy: Remember that differentiability is a stricter condition than continuity. If a function is not continuous at a point, it cannot be differentiable at that point. However, simply being continuous does not guarantee differentiability (as demonstrated by the piecewise function example where LHD $\ne$ RHD at $x=0$).

4. Euler’s Theorem Check: For verifying homogeneity, substitute $x=tx$ and $y=ty$. If $f(tx,ty)=t^{n}f(x,y)$, then the function is homogeneous of degree $n$. This is often faster than rewriting the function into the $\Phi (y/x)$ format. Euler’s theorem ($x{f}_x+y{f}_y=nf$) then provides a quick check or solution shortcut for partial differentiation problems involving homogeneous functions.

5. Curvature Formula Memory Aid: When dealing with the radius of curvature ($\rho$) for explicit functions, remember the structure: $\rho =\frac{(1+(\text{first derivative}{)}^2{)}^{3/2}}{\text{second derivative}}$. The higher the power, the higher the derivative that must be inside (in the numerator, $y_1$; in the denominator, $y_2$).