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 As always, practice and understanding the basic rules of Calculus will help make solving problems like these easier.MATHS :: Lecture 06 ::Differential Calculus(2) It is important to note that the Product Rule applies to any combination of functions where one function is being multiplied by another, not just the examples shown here. In these examples, we can see how the Product Rule can be applied to different types of functions and how it can be used in combination with other rules to find the derivatives of more complex functions. Using the product rule, the derivative of f(x) = x^4 * cos x = (x^4)' * cos x + x^4 * (cos x)' = 4x^3 * cos x - x^4 * sin xįind the derivative of f(x) = (x^2 + 3x + 2) * (e^x + x^2) Using the product rule, the derivative of f(x) = e^x * ln x = (e^x)' * ln x + e^x * (ln x)' = e^x * ln x + e^x * (1/x)įind the derivative of f(x) = (x^2+5x+6) * (2x^2+x) Using the product rule, the derivative of f(x) = x^3 * sin x = (x^3)' * sin x + x^3 * (sin x)' = 3x^2 * sin x + x^3 * cos x It's important to note that the product rule can also be used in combination with other rules, such as the Chain rule, to find the derivatives of more complex functions.įind the derivative of f(x) = x^3 * sin x ![]() ![]() ![]() This rule allows us to find the derivative of a function that is the product of two other functions.įor example, if we have a function f(x) = x^2 and a function g(x) = 3x, we can use the Product Rule to find the derivative of the function h(x) = f(x) * g(x) = x^2 * 3x = 3x^3. ![]() This rule is represented mathematically as: The Product Rule is a rule in Calculus that states that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function.
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