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 \(3x 1\). 1 Form the two possible compositions of f ( x) x and g ( x) 625 x 2 and compute the derivatives. The only deal is, you will have to pay a penalty. In general, if f ( x) and g ( x) are functions, we can compute the derivatives of f ( g ( x)) and g ( f ( x)) in terms of f ( x) and g ( x). So, cover up that \(3x 1\), and pretend it is an \(x\) for a minute. This is a great example of the power of matrix notation. This formula is wonderful because it looks exactly like the formula from single variable calculus. You know by the power rule, that the derivative of \(x^5\) is \(5x^4\). The Sum and Difference rules simply say that the derivative of a sum or difference is the sum or difference of the derivatives. The chain rule tells us that h ( x) f ( g ( x)) g ( x). So, there are two pieces: the \(3x 1\) (the inside function) and taking that to the 5th power (the outside function). 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 In what follows, the functions f f and g g look like lines however, the young mathematician should realize that. We’ll try to understand this geometrically. How the formula for the chain rule works Now we’ll use linear approximations to help explain why the chain rule is true.
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