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Chapter 3: Problem 49

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

Short Answer

Expert verified
Question: Find the derivative of the function $$y=\frac{2 e^{x}+3 e^{-x}}{3}$$. Solution: The derivative of the given function is $$\frac{dy}{dx} = \frac{2}{3}e^{x} - e^{-x}.$$

Step by step solution

01

Identify the terms and simplify the function if possible

The given function is: $$y=\frac{2 e^{x}+3 e^{-x}}{3}$$ Let's see if there is a way to simplify the function. In this case, we can divide each term in the numerator by 3 as follows: $$y = \frac{2}{3}e^{x} + \frac{3}{3}e^{-x}$$ So, the simplified function is: $$y = \frac{2}{3}e^{x} + e^{-x}$$
02

Apply the constant rule

The constant rule states that the derivative of a constant times a function is the constant times the derivative of the function. In this case, we have: $$\frac{d}{dx}(\frac{2}{3}e^{x}) = \frac{2}{3}\frac{d}{dx}(e^{x})$$ $$\frac{d}{dx}(e^{-x}) = \frac{d}{dx}(e^{-x})$$
03

Apply the chain rule and sum rule

The chain rule states that if we have a composite function, its derivative is the product of the derivative of the outer function and the derivative of the inner function. In this case, we have an exponent in the second term, which is a composite function. The sum rule states that the derivative of the sum of two functions is the sum of their derivatives. Applying these rules, we have: $$\frac{d}{dx}(y) = \frac{2}{3}\frac{d}{dx}(e^{x}) + \frac{d}{dx}(e^{-x})$$
04

Calculate the derivative of each term

Now we can determine the derivative of each term in the function. For the first term, we have: $$\frac{d}{dx}(e^{x}) = e^{x}$$ And for the second term, applying the chain rule, we have: $$\frac{d}{dx}(e^{-x}) = e^{-x} \cdot (-1) = -e^{-x}$$
05

Combine the results

Finally, combine the results of the derivatives of each term to compute the overall derivative of the function: $$\frac{d}{dx}(y) = \frac{2}{3}e^{x} - e^{-x}$$
06

Write the final answer

The derivative of the given function is: $$\frac{dy}{dx} = \frac{2}{3}e^{x} - e^{-x}$$

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