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Application Of Derivatives

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APPLICATION OF DERIVATIVES

Derivatives help us understand how things change. Imagine you’re driving a car and you want to know how fast you’re going at any moment. Derivatives tell you the rate of change of your speed.

Derivatives are used to identify maximum or minimum values of functions. This finds application in economics for profit maximization or cost minimization, and in physics for determining paths of least resistance or optimizing efficiency.

Derivatives are incredibly versatile mathematical tools that have numerous applications across different fields. Here are some key areas where derivatives are extensively used:

Rate of Change Calculation
Value Approximation
Tangent and Normal Line Determination
Optimization
Function Behavior Analysis

RATE OF CHANGE

The concept of rate of change is fundamental in mathematics, particularly within the realm of calculus, and it has numerous practical applications across various fields. Essentially, rate of change describes how one quantity varies concerning another. For instance, within calculus, if we have a function y=f(x), the rate of change of with respect to is represented by its derivative $\frac{dy}{dx}$ . This derivative gives us the instantaneous rate of change of at a specific point, indicating how quickly is changing concerning at that precise location.

Rate of change provides insight into how quickly a quantity is changing concerning another. For example, if $y$ represents the position of an object at time $x$ , then $\frac{dy}{dx}$ represents the object’s velocity at time $x$ .

POSITIVE AND NEGATIVE RATES

A positive rate of change indicates an increase in the quantity, while a negative rate of change indicates a decrease. For example, in terms of temperature, a positive rate of change implies a temperature increase, while a negative rate of change suggests a temperature decrease.

AVERAGE RATE OF CHANGE

The average rate of change of a function over an interval $[a, b]$ is calculated as $\frac{f(b) – f(a)}{b-a}$ This represents the change in $y$ over the change in $x$ across the interval $[a, b]$ .

INSTANTANEOUS RATE OF CHANGE

The instantaneous rate of change is the rate at a specific point. It is determined by taking the limit as the interval over which the change is measured approaches zero, which is precisely what the derivative provides.

MAXIMA, MINIMA AND POINT OF INFLECTION

Maxima and minima are critical points on the graph of a function where it either reaches its highest (maxima) or lowest (minima) value within a particular range. These points are essential in optimization problems where one aims to find the maximum profit, minimum cost, or other extreme values. In real-world scenarios, these points could represent the highest or lowest points of efficiency, production, or any measurable quantity.

Points of inflection indicate locations on the graph where the curvature changes. In simpler terms, if you imagine tracing along the graph of a function, at a point of inflection, you would perceive a shift in the direction of curvature. This change can occur from the graph bending upwards to downwards or vice versa. Understanding points of inflection is crucial for accurately sketching the shape of a function’s graph and comprehending its overall behavior.

INCREASING AND DECREASING FUNCTIONS

INCREASING FUNCTIONS

When we say a function is increasing over an interval, it means that as you move from left to right along that interval, the function’s values consistently go up. In simpler terms, if you pick any two points within that interval, the function’s value at the second point will be greater than the value at the first point. This behavior is visually represented by the graph of the function rising as you move from left to right.

For example, let’s take the function $f(x)=x^{2}$ . As $x$ increases from negative to positive values, the corresponding $f (x)$ values also increase. Hence, for $x \geq 0$ , $f(x)=x^{2}$ is an increasing function.

DECREASING FUNCTIONS

Conversely, a function is considered decreasing over an interval if, as you progress from left to right along that interval, the function’s values consistently decrease. In simpler terms, for any two points within that interval, the value of the function at the second point will be lesser than the value at the first point. This trend is depicted by the graph of the function descending as you move from left to right.

For instance, consider the function $f(x)=-x^{2}$ . As $x$ increases from negative to positive values, the corresponding $f (x)$ values decrease. Thus, for $x \geq 0$ , $f(x)=-x^{2}$ is a decreasing function.

TANGENT AND NORMAL TO A CURVE

TANGENT TO A CURVE

A tangent to a curve at a specific point is a straight line that touches the curve precisely at that point, without crossing it. It represents the instantaneous direction of the curve at that point. The slope of the tangent line at any point on the curve is equivalent to the derivative of the curve function at that particular point.

For instance, if we have a curve represented by $y = f (x)$ , then the slope of the tangent line at the point $x_{0},y_{0}$ is given by $f’ (x_{0})$ , where $f’ (x)$ is the derivative of $f (x)$ with respect to $x$ . The equation of the tangent line can be expressed using the point-slope form as $y-y_{0}=f'(x_{0})(x-x_{0})$

TANGENT TO A CURVE

Derivative

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DERIVATIVES

A derivative is a concept used to measure the rate of change of a function concerning one of its independent variables. Visually, it signifies the steepness of the tangent line to the function’s graph at a specific point.

If f(x) represents a function of, then the derivative of $f$ concerning $x$ , denoted as f'(x), $\frac{df}{dx}$

DERIVATIVE FORMULA

We have derivative formulas for algebraic, trigonometric, logarithmic, exponential functions and these formulas are derived from the first principle of differentiation.

POWER RULE FOR DERIVATIVE

f'(x) = nx^{n-1}

If $f(x)=x^{n}$ , where $n$ is a constant, then the derivative $f'(x) = nx^{n-1}$

DERIVATIVES OF TRIG FUNCTIONS

Derivative of sin

\frac{d}{dx}(\sin x) = \cos x

Derivative of cos

$\frac{d}{dx}(\cos x) = -\sin x$

Derivative of tan

$\frac{d}{dx}(\tan x) = \sec^2 x$

Derivative of cot

\frac{d}{dx}(\cot x) = -\csc^2 x

Derivative of sec

$\frac{d}{dx}(\sec x) = \sec x \tan x$

Derivative of cosec

\frac{d}{dx}(\csc x) = -\csc x \cot x

DERIVATIVES OF LOGARITHMIC FUNCTIONS

Derivative of lnx

\frac{d}{dx}(\ln x) = \frac{1}{x}

Derivative of logx

$\frac{d}{dx}(\log_a(x)) = \frac{1}{x\ln(a)}$

DERIVATIVES OF EXPONENTIAL FUNCTIONS

Derivative of e

\frac{d}{dx}(e^x) = e^x

DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS

Derivative of arcsin

\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 – x^2}}

Derivative of arccos

$\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1 – x^2}}$

Derivative of arctan

\frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2}

Derivative of arccsc

$\frac{d}{dx}(\text{arccsc} x) = -\frac{1}{|x|\sqrt{x^2 – 1}}$

Derivative of arcsec

$\frac{d}{dx}(\text{arcsec} x) = \frac{1}{|x|\sqrt{x^2 – 1}}$

Derivative of arccot

$\frac{d}{dx}(\text{arccot} x) = -\frac{1}{1 + x^2}$

BASIC DERIVATIVE RULES

CONSTANT RULE DERIVATIVE

f'(x) = 0

SUM RULE DERIVATIVE

(f + g)'(x) = f'(x) + g'(x)

DIFFERENCE RULE DERIVATIVE

(f - g)'(x) = f'(x) - g'(x)

PRODUCT RULE DERIVATIVE

(fg)'(x) = f'(x)g(x) + f(x)g'(x)

QUOTIENT RULE DERIVATIVE

\left(\frac{f}{g}\right)'(x) = \frac{f'(x)g(x) – f(x)g'(x)}{(g(x))^2}

CHAIN RULE DERIVATIVE

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

IMPLICIT DIFFERENTIATION

Implicit differentiation is a method utilized in calculus to determine the derivative of an implicitly defined function. When a function is defined implicitly, it means that the relationship between the variables is not explicitly stated. Instead, the function is expressed through an equation involving both variables.

The primary steps involved in implicit differentiation are as follows:

1. Differentiate Both Sides

2. Apply Chain Rule

3. Isolate the Derivative

4. Simplify if Necessary

For example, consider the equation of a circle: $x^{2}+y^{2}=r^{2}$ denotes the radius of the circle.

Differentiate Both Sides: Differentiate both sides of the equation with respect to $x$

\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(r^2)

Apply Chain Rule: Since $y$ is a function of $x$ , the chain rule is applied when differentiating $y^{2}$

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

Isolate the Derivative: Solve for $\frac{dy}{dx}$
$\frac{dy}{dx} = -\frac{x}{y}$

PARTIAL DERIVATIVE

Partial derivatives are a fundamental concept in multivariable calculus, enabling us to understand how functions change concerning each of their variables while holding others constant.

In a function $f (x, y)$ of two variables $x$ and ,the partial derivative of $f$ with respect to $x$ , denoted $\frac{\partial f}{\partial x}$ represents how $f$ changes with changes in $x$ , while $y$ remains fixed. Similarly, the partial derivative of $f$ with respect to $y$ , denoted $\frac{\partial f}{\partial y}$ indicates the rate of change of $f$ with changes in $y$ , while $x$ is held constant.

To compute a partial derivative, we differentiate the function with respect to the variable of interest while treating other variables as constants. For instance, for the function $f(x, y) = x^{2} + y^{2}$

The partial derivative $\frac{\partial f}{\partial x}$ is obtained by treating $y$ as constant and differentiating $f$ with respect to $x$ , resulting in $2 x$ .
Similarly, $\frac{\partial f}{\partial y}$ is found by treating $x$ as constant and differentiating $f$ with respect to $y$ , yielding $2 y$

Partial derivatives find wide applications in various fields, including physics, engineering, economics, and optimization problems, where functions depend on multiple variables simultaneously.

PARAMETRIC DERIVATIVES

Parametric derivatives refer to the derivatives of parametric equations, which describe the coordinates of a point in terms of one or more independent variables, often denoted as t.

Given parametric equations x = x(t) and y = y(t), the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$ represent the rates of change of x and y with respect to t , respectively.

For example, for the parametric equations $x(t) = \cos(t)$ and $y(t) = \sin(t)$ , we find:

$\frac{dx}{dt} = \sin(t)$

$\frac{dy}{dt} = \cos(t)$

These derivatives describe how the coordinates x and y change as the parameter t varies, providing valuable insights into the motion or trajectory described by the parametric equations.

Integral

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INTEGRAL

Integration is the calculation of an integral. It involves finding the antiderivative of a function. Integrals in mathematics are used to find various quantities, such as areas, volumes, and displacements.

We can categorize the Integrals into two main types:

Definite Integrals:
Indefinite Integrals:

DEFINITE INTEGRAL

These have limits or boundaries and yield a numerical answer. For example, finding the area under a curve between two points is a definite integral because it involves boundaries.

\int_{a}^{b} f(x) \, dx = f(b) – f(a)

INDEFINITE INTEGRAL

These do not have limits or boundaries and yield an algebraic answer. These involve finding antiderivatives, which means undoing the process of differentiation.

\int f(x) \, dx + C

INTEGRAL FORMULA

BASIC INTEGRAL

\int k \, dx = kx + C

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1$

\int \frac{1}{x} \, dx = \ln|x| + C

TRIGONOMETRIC INTEGRAL

\int \sin(x) \, dx = -\cos(x) + C

$\int \cos(x) \, dx = \sin(x) + C$

$\int \sec^2(x) \, dx = \tan(x) + C$

\int \csc^2(x) \, dx = -\cot(x) + C

$\int \sec(x) \tan(x) \, dx = \sec(x) +C$

\int \csc(x) \cot(x) \, dx = -\csc(x) +C

$\int e^x \, dx = e^x + C$

$\int \frac{1}{x} \, dx = \ln|x| + C$

EXPONENTIAL AND LOGARITHMIC INTEGRAL

$\int e^x \, dx = e^x + C$

$\int \frac{1}{x} \, dx = \ln|x| + C$

\int a^x \, dx = \frac{a^x}{\ln(a)} + C

INVERSE TRIGONOMETRIC INTEGRAL

\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C

$\int \frac{-1}{\sqrt{1-x^2}} \, dx = \arccos(x) + C$

\int \frac{1}{1+x^2} \, dx = \arctan(x) + C

$\int \frac{-1}{x^2 + 1} \, dx = \text{arccot}(x) + C$

$\int \frac{1}{x\sqrt{x^2 -1}} \, dx = \text{arcsec}(x) + C$

$\int \frac{-1}{x\sqrt{x^2 -1}} \, dx = \text{arccsc}(x) + C$