Calculus is perhaps, the most essential and important field of Mathematics, and it has applications in almost every field. However, just like any other sub-field of mathematics, it involves a lot of formulas. So, having a list of calculus formulas in one place is a great accessory.
Specially, in exam days, these lists help students to memorize all the formulas very easily ( We don’t suggest you use them as cheating material).
With that in mind, we have compiled a list of calculus formulas in one article. We hope it helps you.
The list will be categorized according to sub-sections of calculus. Firstly, the basic formulas for differential calculus will be listed, then integral calculus.
These formulas are also sometimes called as laws of calculus.
So, let’s get on with the list of calculus formulas.
Differential Calculus Formulas
In differential calculus, you split up an area into infinitesimal small parts to calculate the rate of change.
The formula is basically derived from the formula of calculating the rate of change of function over some time interval. That formula for average rate of change over an interval is:
The formula above represents the rate of change between interval “x” to “Delta x”.
Ab-Initio Method or First Principle
The ab-initio method or First Principle is the basic formula for calculating derivative of a function.
In formula written above for average rate of change, if the interval is reduced to infinitesimally small interval, then that rate of change is called as derivative.
It is the base calculus formula in all of calculus. It is:
Now, that is the formula for calculating the derivative of a function.
However, in order to simplify the process, derivatives of known functions and some rules in basic mathematical conditions can be memorized to speed up the process of derivation of complex functions.
Rules of Derivatives
These simplifications are:
PRODUCT RULE: For finding derivative of product of two functions. In formula below, two functions are f and g:
POWER RULE: For finding the derivatives, when the function has an exponent.
QUOTIENT RULE: For finding the derivative of quotient of two functions. Again, two functions are going to be f and g:
CHAIN RULE OF DIFFERENTIATION: Chain rule is used to find the derivative of a function which contains another function in it. For example, if y=f(x) is one function and that u in turn equal to u=g(x), then if we are to find the derivative of y with respect to x, then by chain rule:
Derivatives of Trignometric Functions
When remembering the list of calculus formulas, memorizing the results obtained from the ab-initio method for trigonometric functions can really speeds up the derivation process.
They results are:
Formulas for Integral Calculus
Integral calculus is opposite of differential calculus. Here, instead of chopping up a function into pieces to find the rate of change, a small portion of a function is combined together to find the overall result over a range.
“C” in all integration constant represents the constant of integration. As, in differentiation, the rate of change of a function is calculated. Now, that rate of change can be of many functions. C represents a family of functions with that rate of change, that has been integrated. This is called as general solution.
To get individual curves, values of C are used, which are evaluated using a method is beyond scope of this article, but you can learn that here in this article, for finding the particular solution.
Rules of Integration or Formulas for integration are as below:
CONSTANT RULE: To find the integral of a constant “k”:
CONSTANT MULTIPLE RULE: When a constant “k” is being multiplied with function to be integrated:
POWER RULE: When the function to be integrated has an exponent:
For integrating the product of functions, two methods are used, which are called integration by parts and u substution. They are beyond the scope of this article. Follow the link to learn more about them.
Integrals of other know functions like trigonometric etc are below:
So, these was the list of calculus formulas.
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