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Chapter 3: Problem 19

Find \(d y / d x\) for the following functions. $$y=e^{-x} \sin x$$

Short Answer

Expert verified
Question: Find the derivative of the function $$y = e^{-x}\sin x$$ with respect to x. Answer: The derivative of the given function with respect to x is $$\frac{dy}{dx} = e^{-x}(-\sin x + \cos x).$$

Step by step solution

01

Differentiate \(e^{-x}\)

To find the derivative of \(e^{-x}\), we can rewrite it as $$e^{-x} = e^{(-1)x}$$ Now, differentiating \(e^{(-1)x}\) with respect to x, we get: $$\frac{d}{dx}e^{(-1)x} = (-1)e^{(-1)x} = -e^{-x}.$$
02

Differentiate \(\sin x\)

Now, let's find the derivative of \(\sin x\). Using the basic rules of differentiation, we know that the derivative of \(\sin x\) with respect to x is: $$\frac{d}{dx}\sin x = \cos x.$$
03

Apply the product rule

Now it's time to apply the product rule. Recall that we have \(f=e^{-x}\), \(g=\sin x\), \(f'=-e^{-x}\), and \(g'=\cos x\). Using the product rule formula $$(fg)' = f'g + fg',$$ we can find the derivative of the given function: $$\frac{d}{dx}(e^{-x}\sin x) = (-e^{-x})(\sin x) + (e^{-x})(\cos x).$$
04

Simplify the result

Now, let's simplify the result by factoring out the common term \(e^{-x}\): $$\frac{d}{dx}(e^{-x}\sin x) = e^{-x}(-\sin x + \cos x).$$ So, the derivative of the given function with respect to x is: $$\frac{dy}{dx} = e^{-x}(-\sin x + \cos x).$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule
When differentiating a product of two functions, the product rule is a savior. It allows us to easily find the derivative by using a specific formula.
Let's say you have two functions, which are called \(f(x)\) and \(g(x)\). What the product rule tells us is that the derivative of their product \( (fg)' \) can be calculated as:
  • \( f'g + fg' \)
Here, \(f'\) is the derivative of \(f\), and \(g'\) is the derivative of \(g\). In simple terms, you take the derivative of one function and multiply it by the other function, then add the product of the other way around.
This rule is particularly useful because it saves you from manually multiplying the functions together and then differentiating that result, which can be quite complex. Instead, it provides a shortcut that is both time-saving and straightforward. Using this rule often involves differentiating exponential and trigonometric functions, just like in this problem.
Exponential Function Differentiation
Exponential functions, like \(e^x\), have a unique property: their derivative is proportional to the original function. This makes them a popular choice in mathematics.
When you are dealing with \(e^{-x}\), to differentiate this function, remember to apply the rule for exponential differentiation:
  • The derivative of \(e^{u(x)}\) is \(u'(x) e^{u(x)}\).

In our case, \(u(x) = -x\), and differentiating that gives us \(-1\).
Using the formula, the derivative of \(e^{-x}\) becomes \(-e^{-x}\), because:
  • Multiply the derivative of \(-x\) which is \(-1\) with \(e^{-x}\).
This showcases how straightforward yet powerful exponential differentiation rules are. You end up with the exponential term itself, modified by the derivative of the exponent.
Trigonometric Function Differentiation
Trigonometric functions pop up frequently in differentiation problems, and knowing how to handle them smoothly is key.
Differentiating trigonometric functions often involves their basic derivatives:
  • The derivative of \(\sin x\) is \(\cos x\).
  • The derivative of \(\cos x\) is \(-\sin x\).

In this particular task, we wanted the derivative of \(\sin x\), which is \(\cos x\). This straightforward rule helps in simplifying complex expressions into more manageable pieces.
Being conversant with the differentiation of basic trigonometric functions like sine and cosine allows you to quickly apply these rules in larger differentiation problems involving products and chains, making your calculus experience much easier.

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Most popular questions from this chapter

Find \(f^{\prime}(x), f^{\prime \prime}(x),\) and \(f^{\prime \prime \prime}(x)\) \(f(x)=\frac{x^{2}-7 x}{x+1}\)

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Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}(x f(x))\right|_{x=3}$$

An observer is \(20 \mathrm{m}\) above the ground floor of a large hotel atrium looking at a glass-enclosed elevator shaft that is \(20 \mathrm{m}\) horizontally from the observer (see figure). The angle of elevation of the elevator is the angle that the observer's line of sight makes with the horizontal (it may be positive or negative). Assuming that the elevator rises at a rate of \(5 \mathrm{m} / \mathrm{s}\), what is the rate of change of the angle of elevation when the elevator is \(10 \mathrm{m}\) above the ground? When the elevator is \(40 \mathrm{m}\) above the ground?
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