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Practice MCQ

Unit 2 - Notes

MTH165

Unit 2: Differential and Integral Calculus

Part A: Differential Calculus

1. General Rules of Differentiation

Let $u$ and $v$ be differentiable functions of $x$ , and $c$ be a constant.

Constant Rule:
$\frac{d}{dx}(c) = 0$
Scalar Multiple Rule:
$\frac{d}{dx}(cu) = c \frac{du}{dx}$
Sum and Difference Rule:
$\frac{d}{dx}(u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx}$
Product Rule (Leibniz Rule):
$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$
Quotient Rule:
$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}, \quad (v \neq 0)$
Chain Rule (Function of a Function):
If $y = f(u)$ and $u = g(x)$ , then:
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

2. Derivatives of Standard Functions

Algebraic Functions

Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$
Square Root: $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$
Reciprocal: $\frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2}$

Exponential and Logarithmic Functions

Natural Exponential: $\frac{d}{dx}(e^x) = e^x$
General Exponential: $\frac{d}{dx}(a^x) = a^x \ln(a)$
Natural Logarithm: $\frac{d}{dx}(\ln x) = \frac{1}{x}$
General Logarithm: $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$

Trigonometric Functions

$\frac{d}{dx}(\sin x) = \cos x$
$\frac{d}{dx}(\cos x) = -\sin x$
$\frac{d}{dx}(\tan x) = \sec^2 x$
$\frac{d}{dx}(\cot x) = -\csc^2 x$
$\frac{d}{dx}(\sec x) = \sec x \tan x$
$\frac{d}{dx}(\csc x) = -\csc x \cot x$

Inverse Trigonometric Functions

$\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$
$\frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1+x^2}$
$\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}$
$\frac{d}{dx}(\csc^{-1} x) = -\frac{1}{|x|\sqrt{x^2-1}}$

3. Derivatives of Parametric Forms

When variables $x$ and $y$ are both expressed as functions of a third variable $t$ (parameter), i.e., $x = f(t)$ and $y = g(t)$ .

First Derivative:

\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}

Second Derivative:
Note: You must apply the chain rule again regarding $x$ .

\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx}

\frac{d^2y}{dx^2} = \frac{1}{\frac{dx}{dt}} \cdot \frac{d}{dt}\left( \frac{g'(t)}{f'(t)} \right)

4. Derivatives of Implicit Functions

An implicit function is given in the form $f(x, y) = 0$ , where $y$ cannot be easily isolated on one side.

Method:

Differentiate both sides of the equation with respect to $x$ .
Apply the Chain Rule to terms involving $y$ (e.g., $\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}$ ).
Group all terms containing $\frac{dy}{dx}$ on one side.
Solve for $\frac{dy}{dx}$ .

Example:
Given $x^2 + y^2 = a^2$ :

2x + 2y\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}

5. Logarithmic Differentiation

This technique is used for functions of the form $y = [f(x)]^{g(x)}$ or complicated products/quotients.

Procedure:

Take the natural logarithm ( $\ln$ ) of both sides: $\ln y = g(x) \ln(f(x))$ .
Differentiate implicitly with respect to $x$ :
$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left[ g(x) \ln(f(x)) \right]$
Solve for $\frac{dy}{dx}$ :
$\frac{dy}{dx} = y \cdot \left[ \text{derivative of RHS} \right]$
Substitute original $y$ back into the result.

Part B: Integral Calculus

1. Properties of Indefinite Integral

Let $\int f(x) dx = F(x) + C$ .

Differentiation and Integration are inverse processes:
$\frac{d}{dx} \int f(x) dx = f(x)$
$\int f'(x) dx = f(x) + C$
Linearity (Scalar Multiple):
$\int k \cdot f(x) dx = k \int f(x) dx$
Linearity (Sum/Difference):
$\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$

2. Methods of Integration: By Parts

Used to integrate the product of two functions. Based on the product rule for differentiation.

Formula:

\int u \, dv = uv - \int v \, du

Selection of $u$ (ILATE Rule):
To simplify the integral, choose $u$ based on the order of priority in the list below (top has highest priority to be $u$ ):

I - Inverse Trigonometric Functions ( $\sin^{-1}x$ , etc.)
L - Logarithmic Functions ( $\ln x$ , etc.)
A - Algebraic Functions ( $x$ , $x^2$ , etc.)
T - Trigonometric Functions ( $\sin x$ , $\cos x$ , etc.)
E - Exponential Functions ( $e^x$ , $a^x$ )

The remaining part of the integrand becomes $dv$ .

3. Methods of Integration: By Partial Fractions

Used for rational functions $\frac{P(x)}{Q(x)}$ where the degree of $P(x) <$ degree of $Q(x)$ . If the degree is equal or greater, perform polynomial division first.

Types of Denominators $Q(x)$ :

Case 1: Non-repeated Linear Factors
$\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$
Case 2: Repeated Linear Factors
$\frac{P(x)}{(x-a)^2(x-b)} = \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}$
Case 3: Irreducible Quadratic Factors
If $Q(x)$ contains a factor like $(ax^2+bx+c)$ that cannot be factored further:
$\frac{P(x)}{(x-d)(ax^2+bx+c)} = \frac{A}{x-d} + \frac{Bx+C}{ax^2+bx+c}$