Differentiating a PolynomialDate: 04/25/99 at 02:56:24 From: Sarah Loader Subject: Differentiation Differentiate with respect to x: f(x) = x^8 + 3x^5 + (3-5x)^4. Date: 04/26/99 at 15:18:58 From: Doctor Jeff Subject: Re: Differentiation Hello, Sarah, When differentiating a polynomial such as x^8 + 3x^5 + (3-5x)^4, you can differentiate each of the terms one at a time and then add them up to get the derivative of the whole thing. Remember that the derivative of ax^n, where a and n are constants, is nax^(n-1). For example, the derivative of 10x^6 with respect to x is 6*10x^(6-1) = 60x^5 Using this method takes care of the first two terms of your function. The third term requires the use of what is called the Chain Rule. Let's say you are trying to differentiate (2x^2 + 3x + 5)^3 Think of (2x^2 + 3x + 5) as a function g(x). The chain rule states that the derivative of (g(x))^n with respect to x is n*(g(x))^(n-1)*g'(x) [Note that g'(x) is a common notation for the derivative of g(x)] For the derivative of (2x^2 + 3x + 5)^3, we thus get 3*(2x^2 + 3x + 5)^2*(4x + 3) You can use this same method for differentiating the third term of the polynomial in your problem. As a side note, you actually are using the Chain Rule every time you differentiate. If you are taking the derivative of x^n, you can consider x equal to the function g(x). Using the rule, you get der (g(x))^n = n*(g(x))^(n-1)*g'(x) But since g(x) = x and g'(x) = der x = 1, we get n*x^(n-1), which is the procedure you followed in differentiating the first two terms. Hope this helped, Sarah. Good luck with solving the problem. - Doctor Jeff, The Math Forum http://mathforum.org/dr.math/ |
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