Product rule calculus 211/23/2023 If you are taking AP Calculus, you will sometimes see that answer factored a little more as follows: That gets multiplied by the first factor: 18(3x-5)^5(x^2+1)^3. Now, do that same type of process for the derivative of the second multiplied by the first factor.ĭ/dx = 6(3x-5)^5(3) = 18(3x-5)^5 (Remember that Chain Rule!) That gets multiplied by the second factor: 6x(x^2+1)^2(3x-5)^6 Your two factors are (x^2 + 1 )^3 and (3x - 5 )^6 As always, practice and understanding the basic rules of Calculus will help make solving problems like these easier.Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. 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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