Problem 36 - Calculus

Question

Compute \(d y / d x\) by differentiating \(\ln y .\) This is LD: $$ y=e^{-\ln x} $$

Step-by-Step Solution

Answer

The derivative \( \frac{d y}{d x} \) is \( -\frac{1}{x^2} \).

Step 1: Simplify the Expression

The original expression given is \( y = e^{-\ln x} \). We know from the properties of logarithms and exponents that \( e^{-\ln x} \) simplifies to \( \frac{1}{x} \). Therefore, we have \( y = \frac{1}{x} \).

Step 2: Differentiate with Respect to x

We need to find \( \frac{d y}{d x} \) where \( y = \frac{1}{x} \). The derivative of \( \frac{1}{x} \) with respect to \( x \) is \( -\frac{1}{x^2} \). This is obtained by writing \( y = x^{-1} \) and using the power rule, which states that \( \frac{d}{dx} x^n = nx^{n-1} \). Thus, \( \frac{d}{dx} x^{-1} = -1 \cdot x^{-2} = -\frac{1}{x^2} \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique used when it’s difficult to differentiate a function directly, especially when the function involves exponents or products of functions. The basic idea is to take the natural logarithm of both sides of the equation first. This often simplifies the equation, making it easier to differentiate. To start, differentiate the equation with respect to the independent variable, using properties of logarithms to your advantage. The derivative of the natural logarithm of a function, \ln(y), is \(1/y \cdot dy/dx\), which can be rearranged to isolate \frac{dy}{dx}\. This step is particularly helpful when dealing with functions featuring complex exponentials, as logarithms can transform products into sums and powers into products.

Power Rule

The power rule for differentiation is a very common and useful technique. It states if \(y = x^n\), then the derivative \frac{dy}{dx}\ is given by \(n \cdot x^{n-1}\). This makes it very straightforward to differentiate polynomial expressions.
In the exercise, when we wrote \( y = x^{-1} \), this was perfect for applying the power rule. The derivative became \(-1 \cdot x^{-2}\). Remember, you always multiply the exponent by the coefficient in front of \(x\) and then reduce the exponent by one.

Exponents

Exponents are a crucial part of many mathematical functions and equations. They represent how many times a number, called the base, is multiplied by itself. For instance, \( x^{-1} \) is a way of writing \(1/x\).
In differentiation, simplifying expressions with exponents is essential. In this exercise, identifying that \(e^{- ext{ln} x}\) simplified as \(1/x\) is key. This transformation utilized the fact that the exponential function and the natural logarithm are inverses of each other. Recognizing these relationships helps in breaking down and simplifying complex equations.

Derivatives

Derivatives represent the rate of change of a function concerning one of its variables. It's a core concept in calculus and is used to calculate everything from the slope of a curve at a particular point to more complex applications.
In this exercise, after transforming the given function to \(y = x^{-1}\), finding the derivative involved calculating how \(y\) changes as \(x\) changes. The result, \(-1/x^{2}\), indicates how steeply the function descends. Mastering derivatives is fundamental, as it forms the basis for solving more complex calculus problems, optimizing functions, and understanding dynamic systems.