Question

Compute the derivative of the following functions. $$y=\frac{2 e^{x}+3 e^{-x}}{3}$$

Short answer

Question: Find the derivative of the function $$y=\frac{2 e^{x}+3 e^{-x}}{3}$$ with respect to x. Answer: The derivative of the function with respect to x is $$f'(x) = \frac{2 e^{x} + 3 e^{-x}}{3}$$

Step-by-step solution

Differentiate the function with respect to x

To differentiate the function, we will use the rules of derivations. The chain rule will be used in this case as we are differentiating a fraction involving two exponential functions and a constant. Using the chain rule, we have: $$f'(x) = \frac{d}{dx}\left(\frac{2 e^{x}+3 e^{-x}}{3}\right)$$

Apply the chain rule

Applying the chain rule, we first differentiate the numerator with respect to x and then divide by the constant, which in this case is 3: $$f'(x) = \frac{1}{3}\left[\frac{d}{dx}(2 e^{x}+3 e^{-x})\right]$$ Now differentiate the terms in the numerator individually: $$f'(x) = \frac{1}{3}\left[2\frac{d}{dx}(e^{x}) + 3\frac{d}{dx}(e^{-x})\right]$$

Differentiate the exponential functions

Recall that the derivative of an exponential function with respect to x is given by: $$\frac{d}{dx}e^{x} = e^{x}$$ We can now differentiate the exponential function: $$f'(x) = \frac{1}{3}\left[2 e^{x} - 3 e^{-x}(-1)\right]$$ Simplify: $$f'(x) = \frac{1}{3}\left[2 e^{x} + 3 e^{-x}\right]$$ So, the derivative of the given function is: $$\boxed{f'(x) = \frac{2 e^{x} + 3 e^{-x}}{3}}$$

Key concepts

Chain Rule

When it comes to computing derivatives, one of the most versatile and essential tools is the chain rule. It's especially helpful when dealing with composite functions – functions made up of two or more functions. Essentially, the chain rule enables us to differentiate a composition of functions by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

For instance, consider a function defined as the composition of two functions, such as \( h(x) = f(g(x)) \). The chain rule tells us that \( h'(x) = f'(g(x)) \times g'(x)\). It's like peeling an onion; we work from the outer layer to the inner core, differentiating as we go. This method can be applied in a variety of situations where functions are nested within each other.

Exponential Functions

Exponential functions form a significant category of functions with widespread applications in mathematics, science, and engineering. An exponential function is characterized by a constant base raised to a variable exponent, typically expressed as \( f(x) = a^x \), where \( a \) is a positive real number, and \( x \) is the exponent.

These functions exhibit growth or decay at a rate proportional to their current value, which makes them particularly useful for modeling real-world phenomena such as population growth, radioactive decay, and financial investments. Notably, the constant \( e \), which is approximately equal to 2.71828, serves as the base for the natural exponential function, \( e^x \), a prevalent form due to its unique properties in calculus.

Derivatives of Exponential Functions

A distinctive feature of exponential functions, and particularly those with base \( e \), is that they are their own derivatives. In other words, the rate at which these functions grow or decay is directly proportional to their current value. For the natural exponential function \( e^x \), the derivative is simply \( \frac{d}{dx}e^x = e^x \).

Even when the exponent is a more complex function of \( x \) rather than just \( x \) itself, the derivative of the function will involve the original exponential function multiplied by the derivative of the exponent, due to the chain rule. For example, the derivative of \( e^{g(x)} \) would be \( e^{g(x)} \times g'(x)\). These rules are crucial when working with compound interest calculations, growth and decay models, and much more.

Differentiation

Fundamentals of Differentiation

Differentiation is a core concept in calculus that deals with finding the derivative of a function. The derivative represents the rate at which a function's output changes as its input changes. In simpler terms, it measures the slope of the tangent line at any point on the graph of a function.

Differentiation spans various rules and formulas to handle different types of functions and situations, including the power rule for polynomials, the product rule for products of functions, and the quotient rule for ratios of functions. The mastery of differentiation techniques such as these is pivotal for students as they advance in calculus and tackle more complex mathematical problems.