An Overview From Early Days
The universe, where we are dwelling, is in continuous phenomena of variations(derivative). Finding the reasons behind these natural changes have always been the most intriguing desire of humans with thinking minds.
They tend to find the relationship between every new change and its effects. The arithmetic operations like ratios, multiplications, and additions are examples of finding these variations in the numbers system. The problem arose when it was needed to determine a change between the small intervals of time.
Although this question is still held under observation in the naturalist circle, Sir Isaac Newton and Gottfried Leibniz did groundbreaking advancements in this regard. Their work led to the formal syntactical construction of the subject, which we call Calculus. The derivative and Integration are the most seminal pillars which extend the Calculus.
Derivation occupies a central space in the field of applied mathematics. It discusses the rate of change at a certain point. The process of finding this rate of change in any function is called differentiation or derivation.
We can calculate the rate of change by calculating the ratio of change in a dependent variable with respect to the change in an independent variable of a function. In most cases, change in the independent variable and change in a dependent variable is denoted by Δx and Δy respectively.
This process might sound similar to that of estimating the average slope between two points. If it is the case then you are heading in the right direction.
The derivation is all about slope because the slope is itself an instant change in the curve.
Basic Derivative Rules
The derivative is a great large concept to understand. It is a very difficult and complex process to solve such a small instant to calculate the derivative. But the scientists and mathematicians had done a great effort for us.
They had provided some formal methods and rules through which we can calculate any type of derivative easily. These are a certain number of rules in derivation that lead to finding the rate of change in any function. Let’s dive into them, and take an overview of how these concepts are working
The Constant Rule: If there exists a constant integer in the way of derivation, it simply says that there is no curve and a point lies on the horizontal axis. Thus, it will produce zero. So the constant rule will be represented as:
f(x)= c will become f’(x)=0
The Power Rule: In the case of variables with power, the exponent in the power will take at the start of the variable, and the power of the variable (exponent) will be reduced by one degree.
f(x)= x4 will become f’(x)= 4x3
The Constant Multiple Rule: If the function begins from a coefficient then this rule states that it would remain there and you should differentiate the rest.
f(x)= 4x2 will become f’(x)= 4(2x)= 8x
The Sum Rule: If the function contains an addition between them, you should differentiate them separately. the derivative of both terms will solve separately.
f(x)= x2 + 2x will become f’(x)= 2x + 2
The Difference Rule: Just like the sum rule, in case of difference you should follow the same pattern. The terms which are separated by a minus will be treated separately for getting
f(x)= x2 – 2x will become f’(x)= 2x – 2
Integration as its literal meaning conveys is the process of summation and adding up the components. So the integration is the process of summing up small pieces to form a large area. But it is not as much easy as we defined in its definition.
Formally, it is the inverse of differentiation and it helps in determining the volume, and area under the curve. In this way, It is also called the anti-derivatives or anti-differentiation.
Related with Derivative
As derivation helps us in finding the rate of change between two points on a curve, parallelly but in a different manner, the Definite integral calculator helps us to measure the extent of change occurring between those points. Graphically, the derivation deals with the projection of certain points on their respective axis while the integration is the shaded area below these points of a curve.
In a bid to achieve accuracy, in integration, the shaded region becomes fragmentized, and adding these to each rectangular strip gives the area of that region.
Due to its nature of summation, integration denotes by ∫.
In case of given points a, and b,
a∫ f(x) dx
Like differentiation, there are also some methods to solve integration. These integration n rules are listed below.
Multiplication by Constant: The constant will remain untouched and integrate the rest.
∫cf(x) dx will become c∫f(x)
Power Rule: The power of function in integration will be treated specially, i.e.,
∫xn dx will become x n+1/n+1 +c
Sum Rule: The sum rule pattern will be the same as it was in the derivation.
∫(h+g)dx= ∫h dx + ∫g dx + c
Difference Rule: The difference rule pattern will also be the same as it was in the derivation.
∫(h-g)dx= ∫h dx – ∫g dx + c
For learning and solving equations of integral and derivatives on runtime without spending a lot of time, one can use online free tools like this online differentiation calculator and Leibniz notation calculator links .
However, there are also millions of other tools like Calculatored, Symbolab, and many others like that which are the precise tools for doing online mathematical calculations easily and precisely.
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In short, derivatives and integration are the core concepts of calculus. without these concepts, calculus is almost nothing. It took a long pause to discover the concept of integration after the concept of integration but besides that interval, these concepts are related to each other and just treated as opposite to each other in calculations.