Reduced to a familiar form to differentiate.ĮxampleBegin() Example : Differentiate the function. If the variable x occurs in the exponent then by taking logarithm it is.Taking logarithm so that the product is converted into a sum. If the function appears as a product of many simple functions then by.Such process is called Logarithmic differentiation. We differentiated a y to get to that point and so we needed to tack a derivative on.įor some problems, first by taking logarithms and then differentiating, Notice the derivative tacked onto the secant. Here we’ve got two product rules to deal with this time. The algebra in these can be quite messy so be careful with that. Remember that very time we differentiate a y we also multiply that term by since we are just using the chain rule. Notice that when we differentiated the y term we used the chain rule.įirst differentiate both sides with respect to x and notice that the first time on left side will be a product rule. In this example we really are going to need to do implicit differentiation of x and write y as y(x). Note as well that the first term will be a product rule.Įxample: Find for the following function. So, just differentiate as normal and tack on an appropriate derivative at each step. Implicit differentiation is especially useful when y’(x)is needed, but it is difficult or inconvenient to solve for y in terms of x.Įxample: Differentiate the following function with respect to x A function is said to be explicit when one variable can be expressed completely in terms of the other variable.įor example, y = x3 + 2x2 + 3x + 1 is an Explicit functionįor example, the implicit equation xy=1 can be solved by differentiating implicitly gives If the variables x and y are related with each other such that f (x, y) = 0 then it is called Implicit function. If G = at2+b sin t +5 is the growth function function the growth rate and relative growth rate. The relative growth rate i.e defined as the absolute growth rate divided by the totalĭry matter production and is denoted by RGR. The derivative is the growth rate (or) the absolute growth rate gr=. Here t is the independent variable and w is the dependent variable. Growth is a function of time t and is denoted by W=g(t) it is called a growth function. The growth of the plant is usually measured in terms of dry mater production and as denoted by W. (the demand function is generally known as Average revenue function). The demand function for a commodity is P= (a - bQ). If the total cost function is C = Q3 - 3Q2 + 15Q. Then the marginal revenue denoted by MR is given byġ. The total revenue function TR is the product of quantity demanded Q and the price P per unit of that commodity then TR = Q.P = f(Q) Similarly if U = u(x) is the utility function of the commodity x then Let us assume that the total cost C is represented as a function total output q. There are a number of related results that also go under the name of "chain rules." For example, if y=f(u) u=g(v), and v=h(x),ĭifferentiate the following with respect to x If y is a function of u ie y = f(u) and u is a function of x ie u = g(x) then y is related to x through the intermediate function u ie y =f(g(x) )įurthermore, let y=f(g(x)) and u=g(x), then MATHS :: Lecture 06 ::Differential Calculus(2)
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