![]() ![]() All of these are composite functions and for each of these, the chain rule would be the best approach to finding the derivative. Since it was actually not just an \(x\), you will have to multiply by the derivative of the \(3x+1\). The only deal is, you will have to pay a penalty. So, cover up that \(3x + 1\), and pretend it is an \(x\) for a minute. You know by the power rule, that the derivative of \(x^5\) is \(5x^4\). So, there are two pieces: the \(3x + 1\) (the inside function) and taking that to the 5th power (the outside function). Examples Example: What is the derivative of x2+x3 The Sum Rule says: the derivative of f + g f + g Example: What is ddv(v3v4) The Difference Rule. ![]() In this example, there is a function \(3x+1\) that is being taken to the 5th power. Exampleįind the derivative of \(f(x) = (3x + 1)^5\). From there, it is just about going along with the formula. Examples using the chain ruleĪs we apply the chain rule, we will always focus on figuring out what the “outside” and “inside” functions are first. In other words, you are finding the derivative of \(f(x)\) by finding the derivative of its pieces. Using the chain rule, if you want to find the derivative of the main function \(f(x)\), you can do this by taking the derivative of the outside function \(g\) and then multiplying it by the derivative of the inside function \(h\). You can think of \(g\) as the “outside function” and \(h\) as the “inside function”. The main function \(f(x)\) is formed by plugging \(h(x)\) into the function \(g\). This looks complicated, so let’s break it down. The chain rule says that if \(h\) and \(g\) are functions and \(f(x) = g(h(x))\), then How the formula for the chain rule works How the formula for the chain rule works.A surprising number of functions can be thought of as composite and the chain rule can be applied to all of them. An example of one of these types of functions is \(f(x) = (1 + x)^2\) which is formed by taking the function \(1+x\) and plugging it into the function \(x^2\). The chain rule is a method for finding the derivative of composite functions, or functions that are made by combining one or more functions.
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