Question

If \(\mathrm{e}^{\mathrm{y}}=\left(\mathrm{e}^{2} / \mathrm{x}^{2}\right)\) and \(\left[\left(\mathrm{d}^{2} \mathrm{y}\right) /\left(\mathrm{d} \mathrm{x}^{2}\right)\right]=\left(\mathrm{A} / \mathrm{x}^{2}\right)\) the \(\mathrm{A}=\) (a) \(-2\) (b) \((1 / 2)\) (c) 2 (d) \((1 / 3)\)

Short answer

The value of A is 2 (option c).

Step-by-step solution

Rewrite the given equation in a simpler form

Rewrite \(e^y = \frac{e^2}{x^2}\) as: \(e^y = e^2 x^{-2}\)

Find the first derivative(dy/dx)

Take the natural logarithm of both sides to simplify the differentiation process: \(y = 2 - 2 \ln{x}\) Now differentiate with respect to x: \(\frac{dy}{dx} = -2 \frac{1}{x}\)

Find the second derivative (d^2y/dx^2)

Differentiate the first derivative with respect to x: \(\frac{d^2y}{dx^2} = 2 \frac{1}{x^2}\)

Compare the second derivative to the given equation

The second derivative is: \(\frac{d^2y}{dx^2} = 2 \frac{1}{x^2}\) Compare this to the given equation (A/x^2): \(\left[\left(\frac{d^2y}{dx^2}\right)\right] = \frac{A}{x^2}\) Thus, we can see that A = 2. Therefore, the correct answer is (c) 2.

Key concepts

Exponential Functions

Exponential functions are a fundamental concept in mathematics, characterized by the equation of the form f(x) = a^x, where a is a positive constant referred to as the base. The key feature of exponential functions is that the rate of change of the function is directly proportional to the value of the function itself.

For example, in our original exercise, the function e^y = e^2 / x^2 is an exponential function with base e, which is the mathematical constant approximately equal to 2.71828. This base of e is particularly important because it leads to properties that are widely used in calculus, such as the natural logarithm which was applied in the solution.

When dealing with these functions, it's crucial to understand how to manipulate expressions involving exponential terms and the implications of the exponential growth or decay depending on the context of the problem.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the inverse operation of taking an exponent with base e. It answers the question: 'To what power must we raise e to obtain x?' This function is integral to solving various calculus problems where exponential functions are present.

In the solution provided, taking the natural logarithm of both sides simplified the differentiation of the exponential equation, transforming it into y = 2 - 2ln(x). This step is a fundamental technique in calculus to linearize exponential equations, making them more manageable and allowing us to apply differentiation rules as done in the next steps of the problem.

Differentiation

Differentiation is a chief operation in calculus which computes the rate of change of a function at any given point. The derivative of a function at a point essentially gives the slope of the tangent line at that point on the curve of the function.

In the context of our exercise, differentiation was applied twice. The first derivative (dy/dx) was determined to find the rate of change of y with respect to x, resulting in -2/x. Subsequently, the second derivative (d^2y/dx^2) was found by differentiating dy/dx with respect to x again, giving 2/x^2. This second derivative is significant as it represents the curvature of the original function's graph, often referred to as the function's 'concavity'.

The second derivative can indicate points of inflection and is critical in understanding the behavior of functions beyond just their instantaneous rate of change.