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The Product Rule:

We use the Product Rule when we have products of two or more functions. In case of three functions, we take any two functions as as and differentiate the third as so on hence, forming three terms in the sum.

There is also a quotient rule for derivatives of functions in the form u/v but, we will stick with the product rule for the form u/v by treating (1/v) as a function of x.
But we should be familiar with the quotient rule too. According to the quotient rule:

Now, if we are dealing with composite functions, we need to use the chain rule to find their derivatives which is stated below:

Now, using the above rules and the table of derivatives in the previous post, we can find the derivative of any function and combination of functions (applying the rules as needed).

List of Derivatives of the basic functions:

All these can be derived using The first principle. However, we do need to get (very) familiar with these.
To learn the basic derivatives, Practice as many questions as you can. First do the questions referring to the table here & then without it.

So, lets see how we can apply the first principle to differentiate functions.
We'll start with the basic functions, lets take the square function first.

So the derivative of the square function is 2x which is also the slope of the tangent at any point on the curve of the function. So, for a point, x=a, the slope of the tangent would be 2a(putting x=a in 2x) And, since we know the slope of this tangent at a, we can also find its equation. Hence, we can use derivatives to find equations of tangents which is an important Application Of The Derivative.

Now, lets differentiate another function using this principle. We'll go for the rectangular hyperbola this time.


Now, lets find out a generalization for such functions(polynomial/rational & irrational).

We considered the value (phi) to collectively consist of all the higher powers of the change in x as the terms with higher powers would eventually cancel out while solving the limit.
Hence, this is the actual method for finding derivatives of functions - the first principle. However, in practice, we use properties of derivatives and the basic derivatives of the most common functions to find the derivatives of bigger and more complex functions. First, we must be aware of the derivatives of the major and most common functions.

In the previuos topic, we found out the slope of the tangent which was the derivative of the function, we had actually found something called the first principle of calculus!

So, the thing in the red box there is the first principle which we will use to find the derivatives of a function. This process, of finding derivatives(notation:dy/dx) or the differential coefficients of functions may be called Differentiation & we are said to differentiate functions while finding the derivatives.

The function y=f(x) in green, is associated with two lines in the above diagram. The line there in blue cuts the function at two points while the line in orange touches the curve at a single point. With some of our previus knowledge, we would say that the line in orange is the tangent line as it touches the curve at one single point. The statement IS ture, but does it mean that the tangent cannot cut the curve furthur at another point?
Well, the figure below answers the question.

Here, we see that the tangents at P & Q do intersect the curve of the sine function again at other points. We also notice that the curve has the same nature between point P & the point 3pi/2 & the tangent intersects the curve on the left of 3pi/2, ie when the curve has changed its nature! We find the same observation for the tangent at Q. So, for any curve, the tangent may intersect it again at a point provided the curve has changed its nature atleast once between the new point of intersection & the point of contact of the tangent.

So, that was about the orange(tangent) line. Lets talk about the blue line. Have you seen something like this before? Sure you have. Have a look at the image below.

So, this line in red on the circle, and in blue on the function, is called the secant line. Now, lets try to find ut the slope of this secant line.

So, the slope comes out to be (the change in y)/(the change in x) & here, the change in y refers to the difference between the value of the function at A and its value at B, which is:

Hence, The slope of the secant line can be written as:

Now, we need to get the orange line into the picture, keeping the blue line in mind.

Now, Lets take our observation a step furthur. You can observe in the figure below, that when the above happens, ie when A gets closer to B, the change in x goes on decreasing and finally, tends to zero when the slopes tend to be equal, or when A & B tend to coincide.

So, we basically have limiting values here which calls for the application of limits.
And that limiting value there folks, is the derivative!

Graphical Transformations help us to plot any function in calculus.
These transformations can easily be observed by changing the values of the independent variable x.

Before we study differential calculus, it is important to understand the concept of functions and their graphs. This is a major pre-requisite before any Calculus course often dealt with in a separate course called Pre-Calculus.

The Concept Of Functions, Domain & Range

Lets say we have a circle. The area of this circle depends on its radius. Hence, we have an example where a quantity depends on another. Thats exactly what the concept of a function is. Here, the area depends on the radius & is said to be ‘a function of the radius’. The area, hence, is also called the ‘dependant variable‘ while the radius being independent is called the ‘independant variable‘. Now, depending on whatever value of radius (the input) is, there is a corresponding value of the area(the output).

Every single machine in the world, works on the concept of a function. We give the machine an input & it gives us an output. Lets take a washing machine for example, we give it dirty clothes(the input) & it gives us clean clothes(the output). But, what if we leave our cell phone in our pants while giving it to the machine. It will not execute the function. You will get an output for the clean pants BUT NOT for the cellphone even though they are together.

This means that the washing machine is not programmed to take in the cell phone as an input or we say, the cellphone does not fall into the domain of the machine. Hence, the domain of a function consists of all the values that can be given to the machine as the input & the range are the values of all the ossible outputs obtained from the machine.

Mathematically, a function is described as y=f(x) where x is the input or the independant variable & y is the output or the dependant variable as explained above. Hence, all possible values of x form the Domain of the function & the corresponding output values of y form its Range.

Here is a table where functions are classified into 3 main categories based on their nature.

Algebraic Functions:

Modulus Function:

Fractional Part & Greatest Integer Function:

Signum Function:

Trigonometric Functions - Sine & Cosine Functions

Trigonometric Functions - Tangent & Cotangent Functions

Keeping these basic functions & their graphs in mind, we will move on to Graphical Transformations

Dried Beans...CALCULUS. Woo...Well, its basically pretty conceptual & if you follow the ten step that follow, you'll be in a better position to ace calculus!

Do remember that these are steps and not ways to ace calculus. So you'll need to go step-by-step. You can move on to the next step only after the execution of the first(mathematicians are programmers).

1. Ace PreCalculus Your precalculus must be very strong if you want to move on to Calculus.

To strengthen PreCalculus, you'll need to master:

• Algebra
• Trigonometry
• Geometry

(including equations & inequalities, trigonometric identities, conics, the binomial theorem, sequences & series.)

If you're planning to take up multivariable calculus as well, you also need to build up concepts of:

• Vectors
• Parametric coordinates & equations
• Matrices

as a part of precalculus itself.

Now, once you are able to see functions all around you...you're ready to go!

3.Understand the Basic Principles Yes, its now time to know what you are trying to ace...Calculus. To understand the basic principles of Calculus, you jest need to ask yourself three questions: What. Who. Why....And the answer is the same... Mr. Newton & Mr. Leibniz gave us: On your left is what we call the derivative(represented by d/dx or y' or f') which gives the slope of the tangent to the function at a point. Say you have the function f(x), then the slope of f(x) at x=a would be f'(a).

On the right is the integral of the function between two points(represented by the two lines without arrow heads) which gives the area under the curve of the function between these two points.

All the x's corresponding to these points(slope or derivative=0) are the points of extrema & the corresponding y's, the maximum & minimum values of the function. C) The third thing derivatives help us with is determining the nature of a function or curve. We can find the intervals in which the function increases or decreses. Take a look at the image below. You'll observe that the slope of the tangent is positive whenever the function increases while the slope is negative when the function decreases. Yeah, I know exactly how you feel!