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Chapter 3: Problem 48

Compute the derivative of the following functions. $$f(x)=(1-2 x) e^{-x}$$

Short Answer

Expert verified
Question: Find the derivative of the function $$f(x) = (1-2x)e^{-x}$$. Answer: The derivative of the given function is $$f'(x) = e^{-x}(2x - 1)$$.

Step by step solution

01

Differentiate g(x)

To distinguish g(x) = 1-2x, use the power rule. The derivative of this function is g'(x)=-2.
02

Differentiate h(x)

To compute the derivative of h(x) = e^(-x), first, let's differentiate the outer function with respect to x, which is e^(u) where u = -x. The derivative of e^(u) is simply e^(u). Now, differentiate u with respect to x, which is -1. Then, apply the chain rule to find h'(x), which is h'(x) = ((-1)e^(-x) = -e^(-x).
03

Apply the product rule

Now that we have found the derivatives of g(x) and h(x), it's time to apply the product rule. Given f(x) = g(x)h(x), f'(x) = g'(x)h(x) + g(x)h'(x).
04

Calculate f'(x)

Substitute the derivatives g'(x) and h'(x) we found, and the original functions g(x) and h(x) into the product rule expression: $$f'(x) = g'(x)h(x) + g(x)h'(x) = -2e^{-x} + (1-2x)(-e^{-x})$$
05

Simplify f'(x)

The derivative can be simplified by factoring out the common term e^(-x): $$f'(x) = e^{-x}(-2 + (-1 + 2x)) = e^{-x}(2x - 1)$$ The derivative of the function is $$f'(x) = e^{-x}(2x - 1)$$.

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Most popular questions from this chapter

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals. . A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Use implicit differentiation if needed to find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=c x^{2} ; x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants

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?

Proof by induction: derivative of \(e^{k x}\) for positive integers \(k\) Proof by induction is a method in which one begins by showing that a statement, which involves positive integers, is true for a particular value (usually \(k=1\) ). In the second step, the statement is assumed to be true for \(k=n\), and the statement is proved for \(k=n+1,\) which concludes the proof. a. Show that \(\frac{d}{d x}\left(e^{k x}\right)=k e^{k x}\) for \(k=1\) b. Assume the rule is true for \(k=n\) (that is, assume \(\left.\frac{d}{d x}\left(e^{n x}\right)=n e^{n x}\right),\) and show this implies that the rule is true for \(k=n+1 .\) (Hint: Write \(e^{(n+1) x}\) as the product of two functions, and use the Product Rule.)

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=\frac{x}{x+5}$$

An observer stands \(20 \mathrm{m}\) from the bottom of a 10 -m-tall Ferris wheel on a line that is perpendicular to the face of the Ferris wheel. The wheel revolves at a rate of \(\pi\) rad/min and the observer's line of sight with a specific seat on the wheel makes an angle \(\theta\) with the ground (see figure). Forty seconds after that seat leaves the lowest point on the wheel, what is the rate of change of \(\theta ?\) Assume the observer's eyes are level with the bottom of the wheel.
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