What is the difference between differentiating and integrating

Formula 3 Here we have the exponent as x and the base of the expression can be 'e' or any other numberic value represented by 'a'. Formula 4 The differentiation of a logarithmic function results in a unit faction with this function in the denominator, and the integration of this same unit fraction results in the logarithmic function.

Formula 5 The differentiation and integration of trigonometric functions are complementary to each other. Futher in the reverse process of integration the negative sign is passed on to the answer. Formula 6 Inverse trigonometric functions has a similar differentiation and integration process as the basic trigonometric functions. Also here too for integration, the negative sign is passed on to the answer. Formula 7 The following formulas of differentiation and integration can be applied for the product of two functions.

In another term, differentiation forms an algebraic expression that helps in the calculation the gradient of a curve at given point. It is important to highlight that curves have their slopes varying at given point unlike straight lines, which have the same gradient all through.

Integration is a term used in calculus to refer to the formula and the procedure of calculating the area under the curve. It is worth noting that the graph must be under a curve, which results to the formation of an integral part, which is difficult to find the area unlike other shapes like circles, squares, and rectangles, which are easier to calculate their areas. Integration and differentiation can be primarily be differentiated in the way the two concepts are applied and their ultimate results.

They are used to arrive at different answers, which is the fundamental difference. Differentiation is used in calculating the gradient of the curve.

Nonlinear curves have different slopes at any given point, which makes it difficult to determine their gradients. The algebraic expression used to determine the change incurred from one point to another with a unit is referred to as differentiation.

On the other hand, integration is an algebraic expression used in calculating the area under the curve because it is not a perfect shape after which area can easily be calculated. Differentiation and Integration algebraic functions are directly opposite of one another, specifically in their application. If one performs integration, he or she is said to be showing the opposite of differentiation while if one performs differentiation, he or she is performing opposite of integration. For example, integration and differentiation form a relationship that is similarly depicted when one performs the square of a number and then finds the square root of the result.

Therefore, if one wants to find the opposite of an integrated number, he or she will be required to perform the differentiation of the same number. Simply, integration is the reverse process of differentiation and vice versa. In real life scenarios, integration and differentiation have been found to be applied differently to each concept used in providing different results. You can still prove a version of u-sub for absolutely continuous functions, but it takes more work.

You still need to check continuity at each point, but you can basically check the points one-by-one. But in any case, every absolutely continuous function is continuous, but plenty of continuous functions are not absolutely continuous. Indeed I would guess "most" continuous functions are not absolutely continuous.

Isn't that an asymmetry? That is, a function and its inverse should be "asymmetric until proven symmetric" rather than the other way around. Nobody ever promised that the inverse action should be as easy as the original one.

And the definition of preimage seems a little arbitrary to me, but perhaps I am mistaken Let me illustrate. Suppose you prepare from each decimal number the sum of its digits: from you get 6, and so on. This is absolutely routine. But now you ask, which numbers give you 6, the preimages of 6? Looks more challenging, isn't it? Finding preimages is not a map, because for one object you may get several. And what if you are asked to find minimal of those preimages?

This weird task is in the spirit of integration problems of traditional calculus: you are asked to find a very special form of answer. This follows from Rolle's Theorem. Denis Serre Denis Serre 45k 8 8 gold badges silver badges bronze badges. Can you perhaps give an example of "Somehow, integration is not the inverse of differentiation.

I don't have a specific example in mind right now. Dirichlet function, Wiener process etc. I wish I could vote this answer up a second time! But I want to elaborate a little also on his first answer: What are elementary functions? Answer to the comment: From elementary functions maybe better, functions with names we construct other functions by composition, etc. Peter Michor Peter Michor 24k 1 1 gold badge 56 56 silver badges bronze badges.

You seem to be saying that with respect to the right "basis" integration could be made routine, behaving in a similar way to how differentiation behaves on elementary functions. This question cannot be answered by showing how easy is differentiation and how hard or even hard to define integration is.

Specially if the functions to be treated are written already in ways that are easy for the derivative to treat. In a sense, we are giving assuming the input of the derivative already in a way that it is easy to compute. Deane Yang Deane Yang True, linear functions are simple, but the linearization we perform when differentiating varies from point to point in a nonlinear way, and it's the entire derivative we're interested in here rather than just the derivative at a point.

This is particularly clear if we look at functions of more than one variable, when the derivative is a decidedly more complicated object than the original function. Somehow linearization is formally algebraically a nice operator, whereas reconstructing a nonlinear function from its linearization is not.

But otherwise I can understand your skepticism. On the other hand I thought and said it above that differentiation is also global in a way because you are trying to find a function which satisfies the whole domain and not only one point. Now Gowers seems to support this point. Isn't this a contradiction? The whole matter gets more and more mysterious at least to me I'll add a few more vague comments to my answer when I get some time.

Michael Renardy Michael Renardy Let me see where it takes us. It is probably too verbose since I am writing it as I think the question. Maybe later I can shorten it and correct errors. If you are hurried it is safe to scroll down to the last part, where there is what I think is an answer to the OP's question.

I want to concentrate on differentiation and integration as symbolic operations. Claim: Integration is, at least, as hard as derivation. Ryan Reich Ryan Reich 6, 4 4 gold badges 34 34 silver badges 52 52 bronze badges.

Thierry Zell Thierry Zell 4, 2 2 gold badges 43 43 silver badges 57 57 bronze badges. Formal calculus on polynomials is certainly equally easy for either operation. Likewise finite Fourier series.

In both cases, we can apply linearity and consult a finite table of derivatives or antiderivatives. However, composition usually produces too many functions to admit only a finite table as sufficient, so this method doesn't go very far unless you want to use infinite series, where everything becomes truly mechanical except for figuring out which closed-form function you have.

I'm pretty sure that you can make a good argument why integration is inverse, but right now the only thing I can think of is the non-uniqueness. Indeed, I would have to agree with them! The computation only really becomes simple when you apply the fundamental theorem of calculus. Best Regards, Juan I. Juan I. Perotti Juan I. Perotti 21 4 4 bronze badges. Todd Rowland Todd Rowland 19 1 1 bronze badge.

The only thing that differs is the representation custom made for people which differs in its complexity? I think that only shifts the problem: Why is it that for symbolic differentiation representation is easy and for integration not? The second half of my post meant to address the human question. The CA analogy, where differentiation is like a step in a CA, shifts the problem to something easier to visualize. That reversing a CA is harder than computing its evolution is something one can see in the wild evolution of rule One of the adhoc methods of integration is knowing what terms to include, and doing this is similar to reversing a CA, both in the cases where it is easy and where it is hard.

Italo Cipriano Italo Cipriano 1, 2 2 gold badges 14 14 silver badges 25 25 bronze badges. Featured on Meta. Main Differences Between Differentiation and Integration. Differentiation is used to calculate the gradient of a curve. It is used to find out the instant rates of change from one point to another.

Differentiation is used to calculate instant velocity. It is also used to find whether a function is increasing or decreasing. Integration is used to calculate the area of curved surfaces. It is also used to calculate the volume of objects. Differentiation is used to calculate the speed of the function as it calculates instant velocity.

Integration is used to calculate the distance covered by any function as it calculates the area under the curve.