Calculus is one of the most important and vast topics of mathematics. It has a lot of applications in real life, because of which it is important to master this topic. There are many concepts in calculus that are difficult to crack. We will discuss both calculus and a few concepts, such as product rule in this article.
The major two operations that one can relate to calculus is differentiation and integration. Differentiation is about breaking into small, whereas integration is about adding small quantities. Integration is both definite and indefinite. If there are limits given to us between which we have to integrate, then the integration is said to be definite. On the other hand, if limits are not given then it is an indefinite integration. In indefinite integration, we always have to mention c in the final result which denotes constant. The graph of a definite integration gives us the area of the integrated expression. There are many different methods of integrating an expression. A few of the methods are partial fraction and integration by parts. Integration of different expressions requires different methods. Like few questions that can be solved only by using parts whereas others can be solved only by a partial fraction. Hence a student should have a good grasp of all the methods of integrating.
Differentiation is another crucial concept of calculus. One key point to note, differentiation of a constant term is always equal to zero. Differentiation in general means the rate of change of a quantity to another quantity. One amazing property of differentiation is that it leads us to find the maximum and minimum value of the expression. The geometric mean of differentiation gives us the slope of the tangent of the curve we are differentiating. There are various methods of differentiating different quantities. If there is a single variable, then its differentiation is done to that variable by decreasing one from the power of the variable and multiplying the original power with the quantity that we got after subtracting the power. Suppose x3 is given to us for differentiating. We will just decrease its power by one that is we will get x2, now multiply the original power that is 3 with x2. Thus, we will get the final result as 3×2 and this is the differentiation of x3.
Now, let us look at the method of differentiation when the expression is given as the product of two or more variables. In such cases, we use the product rule. Suppose we are given the expression f(x)*g(X) that needs to be differentiated to x. We will use the product rule to solve this problem. Product rule states that first differentiate any one expression, let us differentiate f(x) in this example and denote a differentiated term as F(x).
Now multiply it with another variable term that is F(x)*g(x). Now differentiate the other term g(x) in the given example, we will get G(x) and multiply it by the first term, that is G(x)*f(x). Finally add the terms F(x)g(x) and G(x)f(x) , that is we will get the final result as F(x)g(x)+G(x)f(x). We will continue the same process if more terms are given in the product form in the expression that needs to be differentiated.
In the above article, we have tried to discuss calculus, and one of its crucial concepts is product rule. Students must master these topics as they are widely used in higher studies. If any student faces a problem in solving a maths-related topic, then they should take the help of Cuemath. It is an online platform offering quality education to everyone in various countries.