Differential ... For some, this is a beautiful distant, and for others is an incomprehensible word associated with mathematics. But if this is your harsh present, our article will help you learn how to "prepare" differential and with what to "serve".
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Under the differential in mathematics, they understand the linear part of the increment of the function. The concept of differential is inextricably linked with the record of the derivative according to the Labender F '(x 0) \u003d df / dx · x 0. Based on this, the first-order differential for the function f specified on the set X, it has this kind: D x0.f \u003d f '(x 0) · D. x0.x. As you can see, to obtain differential you need to be able to freely find derivatives. Therefore, it will be useful to repeat the rules for calculating the derivatives in order to understand what will happen in the future.
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So, consider differentiation closer on the examples. It is necessary to find the differential function specified in this form: Y \u003d X 3-X. 4. We first find the derivative of the function: y '\u003d (x 3-X. 4) '\u003d (X 3) '- (x 4) '\u003d 3x 2-4x 3. Well, now get differential easier simple: df \u003d (3x 3-4x 3) · DX. Now we have received a differential in the formula formula, in practice, often interests the digital value of the differential at specified specific parameters x and Δx.
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There are cases when the function is expressed implicitly through x. For example, y \u003d x²-y x.. The derivative function has this kind: 2x- (y x.) '. But how to get (y x.) '? This function is called complex and differentiated according to the appropriate rule: DF / DX \u003d DF / DY · DY / DX. In this case: df / dy \u003d x · y x-1, and dy / dx \u003d y '. Now we collect everything together: y '\u003d 2x- (x · y x-1· Y '). We group all the first to the same side: (1 + x · y x-1) · Y '\u003d 2x, and in the end we get: y' \u003d 2x / (1 + x · y x-1) \u003d DY / DX. Based on this, dy \u003d 2x · dx / (1 + x · y x-1). Of course, it is good that such tasks are rarely found. But now you are ready for them.
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In addition to the first-order differentials, there are still top-order differentials. Let's try to find differential for function D / D.(X. 3)·(X. 3–2x. 6–x. 9), which will be the second order differential for F (X). Based on the formula F '(U) \u003d d / d / f (u), where u \u003d f (x), we take U \u003d x 3. We get: D / D (U) · (U-2U 2-u. 3) \u003d (u-2u 2-u. 3) '\u003d 1-4U-3U 2. We return the replacement and get the answer - 1 –x. 3–x. 6, X ≠ 0.
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Assistant in finding differential can also become online service. Naturally, they will not use them on the control or exam. But with an independent verification of the correctness of the solution, its role is difficult to overestimate. In addition to the result, it also shows intermediate solutions, graphs and an indefinite integral of the differential function, as well as the roots of the differential equation. The only drawback is a record in one row of the function when entering, but over time you can get used to this. Well, and naturally, such a service does not cope with complex functions, but everything is simpler, to him on the teeth.
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Practical application The differential finds primarily in physics and economics. Thus, in physics, the differentiation of the tasks associated with the determination of the speed and its derivative - acceleration are solved. And in the economy, the differential is an integral part of the calculation of the efficiency of the enterprise and the fiscal policy of the state, for example, the effect of the financial lever.
This article discusses typical differentiation tasks. The course of higher mathematics of students in universities often contains more tasks for the use of differential in approximate calculations, as well as the search for solutions of differential equations. But the main thing - with a clear understanding of the Azov, you easily deal with all new tasks.
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