Question

Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\). \(f(x)=\ln (1-x)\)

Short answer

Answer: The relationship between a small change in x and the corresponding change in y for the function \(f(x) = \ln(1-x)\) is: \(dy = -\frac{1}{1-x} dx\).

Step-by-step solution

Recall chain rule formula and logarithmic derivatives

To solve this problem, we are going to need to use the chain rule for derivatives, because \(f(x) = \ln(1-x)\) is a composition of two functions. The chain rule states that if \(h(x) = g(f(x))\), then \(h'(x) = g'(f(x)) \cdot f'(x)\). In our case, we have \(g(x) = \ln(x)\), and \(f(x) = (1-x)\). We also recall that the derivative of the natural logarithm is \(g'(x) = \frac{1}{x}\).

Apply chain rule to find the derivative of the given function

Using the chain rule, we can now find the derivative of \(f(x) = \ln(1-x)\) as follows: \(f'(x) = g'(f(x)) \cdot f'(x)\) Now, insert the functions \(g(x)\) and \(g'(x)\) as well as find \(f'(x)\): \(f'(x) = \frac{1}{1-x} \cdot (-1)\) Upon simplification, we get: \(f'(x) = - \frac{1}{1-x}\)

Express the relationship between small changes in x and y

Now that we have found the derivative, we can express the relationship between the small change in x (denoted as \(dx\)) and the corresponding change in y (denoted as \(dy\)) in the form \(dy = f'(x)dx\). Simply substitute \(f'(x)\) into the expression: \(dy = -\frac{1}{1-x} dx\) This is the final answer, which represents the relationship between a small change in x and the corresponding change in y for the function \(f(x) = \ln(1-x)\).

Key concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used in calculus to differentiate functions that can be complex to handle using standard rules of differentiation. This process often simplifies the differentiation of functions that are in the form of one function raised to the power of another, products, or quotients of functions. It exploits the properties of logarithms to turn multiplication into addition, division into subtraction, and exponentiation into multiplication.

For instance, consider a function that is a product of several other functions, logarithmic differentiation allows us to take the natural logarithm of both sides and then differentiate using the sum and difference of the functions. This can be particularly useful when dealing with a function like \(f(x) = \frac{x^2 e^x}{\sin x}\), where using the product and quotient rule directly would be cumbersome.

Derivative of Natural Logarithm

The derivative of the natural logarithm function \( \ln(x) \) is a fundamental concept in calculus. It is defined as \( \frac{1}{x} \), where \(x\) is the input to the function. This derivative is particularly important because it appears frequently in the differentiation of more complex functions, and it forms the basis for more advanced integration and differentiation techniques.

Understanding the derivative of \( \ln(x) \) is crucial when applying the chain rule—especially when the natural logarithm is composed with another function. The rate of change of the natural logarithm reflects how a small change in the input \(x\) results in a proportional change in the function's output, following its inverse relationship with \(x\).

Differential Relationship

In calculus, the differential relationship between variables \(x\) and \(y\) is expressed in terms of their derivatives. Given a function \(y=f(x)\), the derivative \(f'(x)\) represents the rate at which \(y\) changes with respect to a change in \(x\). The term \(dy\), known as the differential of \(y\), is used to approximate the change in \(y\) for a small change in \(x\) denoted by \(dx\).

The differential relationship is therefore written as \(dy = f'(x) dx\). This linear approximation is powerful for understanding how functions behave locally and is foundational to techniques for solving both basic and complex calculus problems.

Function Composition

Function composition involves combining two or more functions to form a new function. It is the process where the output of one function becomes the input of another function. In notation, if \(g\) is a function and \(f\) is another function, then the composition \(g(f(x))\) is read as 'g' of 'f' of 'x'.

This concept is crucial for calculus, particularly when working with the chain rule. The chain rule helps differentiate composite functions by allowing us to find the derivative of complicated composites easily. In the context of the exercise involving \(f(x) = \ln(1-x)\), we see that the function \(f(x)\) is the composition of the natural logarithm and a linear function \(1-x\). Understanding how to work with these composites enables us to apply the chain rule effectively to find derivatives of complex functions.