Final answer:
You can indeed use the chain rule along with the product rule to differentiate the function (1 - ln x)/x^2 by rewriting it as (1 - ln x)(x^-2) and applying the derivatives step by step.
Step-by-step explanation:
Yes, you can use the chain rule to find the derivative of the function (1 - ln x)/x^2 by first rewriting it as (1 - ln x)(x^-2). To differentiate this product, you'll need to use both the product rule and the chain rule.
Here's a step-by-step explanation:
First, apply the product rule which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. In formula form, if u and v are functions of x, then (uv)' = u'v + uv'.
Let u = 1 - ln x and v = x^-2. Compute the derivatives u' and v' separately. The derivative of u with respect to x is -1/x, and the derivative of v with respect to x is -2x^-3 (applying the power rule).
Now, combine these results using the product rule:
(uv)' = u'v + uv' = (-1/x)(x^-2) + (1 - ln x)(-2x^-3).
This simplifies to:
(uv)' = -x^-3 - 2x^-3 + 2(ln x)x^-3.
As a result, the derivative of (1 - ln x)/x^2 with respect to x is -x^-3 (1 + 2 - 2ln x).
This method involves an understanding of both the product rule and chain rule for differentiation.