Implicit differentiation
Implicit differentiation is a technique used to differentiate equations that are not explicitly expressed in terms of one variable. It involves differentiating both sides of an equation with respect to the variable of interest, treating the other variables as functions of it. This method is particularly useful for finding derivatives of equations that are difficult to solve explicitly.
6 Key excerpts on "Implicit differentiation"
eBook - ePub- Adil H. Mouhammed(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...But if the function is written as g(x,y) = 0, then the function is said to be an implicit function. And the differentiation of such a function is called Implicit differentiation. Here, we treat dy/dx as unknown and differentiate the function with respect to x. Having done so, we solve for dy/dx. Example 1: Differentiate xy = 9 implicitly. To differentiate this function implicitly with respect to x, we do the following. Rewrite the function as xy − 9 = 0, and differentiate with respect to x: the derivative of x with respect to x (which is one) times the second variable y, then the derivative of y with respect to x (which is dy/dx) times the first variable x. This gives d y / d x = y + [ dy / dx ] x = 0. Solving for dy/dx, we obtain dy/ dx = − y/x. Example 2: Differentiate x 2 + y 2 − 20 = 0 implicitly with respect to x. Solution: 2x + 2y dy/dx = 0. Thus, dy/dx = − x/y. Example 3: Differentiate xy − y + 4x = 0 implicitly with respect to x. Solution: y + (dy/dx) x − (dy/dx) + 4 = 0. Solving for dy/dx yields d x / d y = − (y + 4) / (x − 1). Example 4 Consumer Equilibrium. A consumer can achieve equilibrium if the indifference curve is tangent to the budget line, that is, if the slope of the indifference curve is equal to the slope of the budget line. This tangency point gives the number of units of commodities x and y that our consumer should buy to maximize his or her utility subject to the amount of income (budget). Suppose the indifference curve equation is X 0.5 Y 0.5 = 15. Let the budget equation be 2x + 3y = M, where x and y are two commodities, and the prices of x and y are $2 and $3, respectively; and x is the total income the consumer has. Given this information, we want to find how many units of x and y our consumer should buy. Solution: Obtain dy/dx, the marginal rate of substitution of x for y, from the indifference equation as well as from the budget equation, then equate the results and solve for x and y...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 7 Applications of Integrals CHAPTER 7 Applications of Integrals 7.1 INTRODUCTION With antidifferentiation skills intact, it is now time to explore the multitude of applications of integrals. Antidifferentiation enables mathematicians, scientists, engineers, economists, and others to “work backwards” from knowledge about changes in quantities to create models of population growth, to determine position functions from knowledge about velocity, to compute average energy use over time, to find marginal cost, and to perform countless other applications. 7.2 DIFFERENTIAL EQUATIONS SEPARABLE DIFFERENTIAL EQUATIONS A differential equation is any equation that relates the instantaneous rate of change of one variable with respect to another to some mathematical combination of the independent and dependent variable. That may sound confusing, but let’s look at a basic example such as The left side of the equation is the “instantaneous rate of change of one variable with respect to another.” The right side of the equation is “some mathematical combination of the independent and dependent variable.” In many cases, the dependent variable may not appear at all in the expression, for example, Differential equations are one of the building blocks of calculus. All differential equations encountered in the first year of calculus are separable. This means, if necessary, all y terms can be grouped with the differential dy and all x terms can be grouped with the differential dx. Of course, a differential equation may be a function of variables other than x and y, but they are all handled the same way: separate the variables and antidifferentiate each side of the equation with respect to the appropriate variable. EXAMPLE 7.1 Solve the differential equation, SOLUTION Separate the variables. becomes dy = (3 x + 5) dx. Now integrate both sides of the equation. Since the constants are arbitrary, they may be combined into one constant on either side of the equation...
eBook - ePub- Bonnie Nguyen, Andrew Wait(Authors)
- 2015(Publication Date)
- Routledge(Publisher)
...In these cases, we can use differential calculus. Differentiation is similar to finding a slope in that it shows how y responds to changes in x (in fact, for a straight line, calculating the slope and differentiating both yield the same answer). The difference is that calculus is concerned with very small changes in x, whereas slope is usually concerned with whole-unit changes. Differentiation is a useful tool because economics is often concerned with marginal changes; in particular, we will make use of it when discussing elasticities (Chapter 10) and determining marginal revenue for a monopolist (Chapter 13). 2.3.1 A simple rule for differentiation For our purposes, we need one rule of differentiation. Take a function y (x) = Ax n. Here, y is written as a function of x. A and n are parameters. By differentiating a function once, we get the first derivative, which we can write as dy / dx or y ′(x). The rule of differentiation that we need is: 2.3 y (x) = A x n ⇒ d y d x = n A x n − 1 This rule can also be applied to each individual additive component of a function, as shown in the example below. Example. Consider a function P (Q) = 10 – Q 2 + 3 Q – 5. We can apply the rule in Equation 2.3 to each individual component (10, –Q 2 and 3 Q – 5) separately. Note that 10 is really 10 Q 0, so after differentiation this becomes 0. Applying the rule to the next two components of the function yields –2 Q and –15 Q – 6 respectively. Thus, the answer is dP / dQ = –2 Q – 15 Q – 6. Like slope, the first derivative of a function tells us how steep a curve is at a particular point. If the first derivative is positive (resp. negative) the curve is rising (resp. falling), and if the first derivative is zero, then the curve is flat. 2.3.2 Finding minima and maxima Differentiation is also a useful way of finding the maximum or minimum of a function. Notice that when a function reaches its maximum or minimum, the slope of the function is zero (that is, the curve is ‘flat’ at that point)...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...10 Differentials, Derivatives, and Partial Derivatives The concept of differential entails a small (tendentially negligible) variation in a variable x, denoted as dx, or a function f { x }, denoted as df { x }; the associated derivative of f { x } with regard to x is nothing but the ratio of said differentials, i.e. df / dx – usually known as Leibnitz’s formulation. In the case of a bivariate function, say, f { x,y }, differentials can be defined for both independent variables, i.e. dx and dy – so partial derivatives will similarly arise, i.e. ∂f / ∂x and ∂f / ∂y ; operator ∂ is equivalent to operator d, except that its use is exclusive to multivariate functions – in that it stresses existence of more than one independent variable. 10.1 Differential In calculus, the differential represents the principal part of the change of a function y =. f { x } – and its definition reads (10.1) where df / dx denotes the derivative of f { x } with regard to x ; it is normally finite, rather than infinitesimal or infinite – yet the precise meaning of variables dx and df depends on the context of application, and the required level of mathematical accuracy. The concept of differential was indeed introduced via an intuitive (or heuristic) definition by Gottfried W. Leibnitz, a German polymath and philosopher of the eighteenth century; its use was widely criticized until Cauchy defined it based on the derivative – which took the central role thereafter, and left dy free for given dx and df / dx as per Eq. (10.1). A graphical representation of differential is conveyed by Fig. 10.1, and the usefulness of differentials to approximate a function becomes clear from inspection thereof; after viewing dy as a small variation in the vertical direction, viz. (10.2) one may retrieve Eq...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...As in these examples, we often start out with information about the derivative of a function without prior knowledge of the equation from which the derivative is derived. The name given to the process of reconstructing an original function from its derivative is integration. In other words, integration is the operation that must be performed to undo the effects of differentiation. We can start to unpack this concept by differentiating the function y = x 2, to achieve the familiar result that d y d x = 2 x. Integration is the reverse of this process, so alongside stating that ‘the derivative of x 2 is 2 x ’, it looks like we should also be able to say ‘the integral of 2 x is x 2 ’. Unfortunately, this picture is a little too simplistic. The true situation is more complex because an infinite number of functions can in fact be differentiated to get a result of 2 x. For example, we would get the same result if instead of y = x 2 we had chosen y = x 2 + 3 or even y = x 2 – 14. In each case, the derivatives of the second terms on the right-hand sides of the equations are zero, because they are all constants. Writing out these derivatives explicitly can emphasize the ambiguity of attempting to undo the effects of differentiation: d y d x (x 2) = 2 x ; d d x (x 2 + 3) = 2 x ; d y d x (x 2 − 14) = 2 x. In any attempt to reverse the process of differentiation, we have no idea what the value of the constant term in the original function may have been. Because of this, it is important to include an unknown constant in our answer for the integral of a function. The unknown constant, often written as c, is called the ‘constant of integration’. This means that the result for the integral of 2 x stated above was only partly correct...
eBook - ePub- Adil H. Mouhammed(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...CHAPTER TWO Derivatives and Applications Economic decisions are based on marginal analysis. For example, the monopolist's best level of output is determined by equating marginal cost and marginal revenue. To find the marginal cost and revenue, total cost and revenue functions must be differentiated with respect to the output level. Similarly, derivatives can be used in many applications in business and economics. For this reason, the rules of differentiation are outlined in this chapter, and many applications are provided. The Concept of Derivative The derivative of a function measures the rate of change of the dependent variable y with respect to the independent variable x--the slope of the function. That is, the derivative indicates the impact of a small change in x on y. For example, suppose the dependent variable y is the quantity supplied by a producer, and x is the price of that product. Mathematically, the function is written as y = f(x). Now, if the price x changes by a very small amount (dx), the quantity supplied will change by a very small amount (dy) as well. These small changes, dx and dy, are called the differential of x and y, respectively. After these changes, the new magnitude of the two variables becomes (y + dy) and (x + dx). And dy/dx is called the derivative of y with respect to x. In other words, dy/dx shows the changes in y per unit change in x. The process of finding the derivative is called the differentiation process. If a given function is a univariate function, such as the above, the following rules of differentiation (Glaister 1984; Chiang 1984; Ostrosky and Koch 1986) are applied: Rule 1: Derivative of a Constant function If y = f(x) = k, where k is a constant, then dy/dx = 0 Example 1: Differentiate y = f(x) = 30. Solution: dy/dx = 0 Example 2: If the fixed cost of a product q is FC = 20, then the derivative of the fixed cost with respect to q is d(FC)/dq = 0...
