/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 51 Tangent lines and general expone... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 51

Tangent lines and general exponential functions.. Find an equation of the line tangent to \(y=x^{\sin x}\) at the point \(x=1\).

Short Answer

Expert verified
Answer: The equation of the tangent line is \(y = (\sin 1)(x-1) + 1^{\sin 1}\).

Step by step solution

01

Find the derivative of y with respect to x

To find the derivative of \(y = x^{\sin x}\) with respect to x, we will need to use the chain rule and power rule. First, we notice that \(y = u^v\) where \(u = x\) and \(v = \sin x\). Let's find the derivatives of \(u\) and \(v\) with respect to x: \(\frac{du}{dx} = \frac{d(x)}{dx} = 1\) \(\frac{dv}{dx} = \frac{d(\sin x)}{dx} = \cos x\) Now, we need to apply the chain rule: \(\frac{dy}{dx} = \frac{du}{dx} \cdot \frac{dy}{du} + \frac{dv}{dx} \cdot \frac{dy}{dv}\) We will find \(\frac{dy}{du}\) and \(\frac{dy}{dv}\) using the power rule: \(\frac{dy}{du} = \frac{d(u^v)}{du} = vu^{v-1}\) \(\frac{dy}{dv} = \frac{d(u^v)}{dv} = u^v \ln u\) Now we plug our values back into the chain rule: \(\frac{dy}{dx} = 1 \cdot (\sin x)(x^{\sin x - 1})+(\cos x)(x^{\sin x}\ln x)\)
02

Evaluate the derivative at x=1

To find the slope of the tangent line, we need to plug x=1 into our derived formula: \(\frac{dy}{dx}|_{x=1} = (\sin 1)(1^{\sin1 - 1})+(\cos 1)(1^{\sin1}\ln 1)\) Since \(\ln 1 = 0\), the second term becomes 0, so : \(\frac{dy}{dx}|_{x=1} = \sin 1\)
03

Find the equation of the tangent line

Now that we have the slope, we can find the equation of the tangent line using the point-slope form of a linear equation: \(y - y_1 = m(x - x_1)\) Where m = slope of the tangent line = \(\sin 1\), \(x_1 = 1\), and \(y_1 = 1^{\sin 1}\). Plugging in the values, \(y - 1^{\sin 1} = (\sin 1)(x - 1)\) So the equation of the line tangent to \(y = x^{\sin x}\) at the point \(x=1\) is: \(y = (\sin 1)(x-1) + 1^{\sin 1}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Derivative of Exponential Functions
Understanding the derivative of exponential functions is essential in calculus, especially when dealing with complex functions involving exponents that are not just constants. In general, if you have an exponential function of the form \(y = a^x\), where \(a\) is a constant, the derivative with respect to \(x\) is \(y' = a^x \ln(a)\). However, the function \(y = x^{\sin x}\) introduces a variable in the exponent, which cannot be differentiated using the standard formula for exponential functions. Instead, we must employ a combination of calculus rules to find its derivative.
Chain Rule Calculus
The chain rule is a fundamental tool in calculus used to differentiate compositions of functions. If you have a function \(y = f(g(x))\), where \(f\) and \(g\) are both differentiable, the chain rule states that the derivative \(y' = f'(g(x))g'(x)\). When applied to the function \(y = x^{\sin x}\), we treat \(x\) and \(\sin x\) as separate functions that compose our main function. The chain rule allows us to differentiate \(y\) with respect to \(x\) by also considering the derivatives of \(x\) and \(\sin x\), thus breaking down a complex derivative into more manageable parts.
Power Rule Calculus
The power rule is another important concept in calculus, which states that for any function \(y = x^n\), where \(n\) is a real number, the derivative is \(y' = nx^{n-1}\). This rule makes it straightforward to find derivatives of functions with power expressions. However, in the case of \(y = x^{\sin x}\), we see that the exponent itself is a function of \(x\). Hence, we combine the power rule with the chain rule to find the derivative, taking into account that the exponent is not constant but rather a variable function dependent on \(x\).
Slope of Tangent Line
The slope of the tangent line to a curve at a particular point is the value of the derivative of the function at that point. This slope indicates the steepness of the line and is essential in constructing the line tangent to the curve. For instance, in the given exercise, we find the slope of the tangent line to \(y = x^{\sin x}\) at \(x = 1\) by evaluating the derivative at that point, resulting in \(m = \sin(1)\). The point-slope form, \(y - y_1 = m(x - x_1)\), is then used to write the equation of the tangent line, using the slope and the coordinates of the given point on the curve.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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?

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\ln (3 x+1)^{4}$$

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}\left(x^{10 x}\right)$$

The total energy in megawatt-hr (MWh) used by a town is given by $$E(t)=400 t+\frac{2400}{\pi} \sin \frac{\pi t}{12},$$ where \(t \geq 0\) is measured in hours, with \(t=0\) corresponding to noon. a. Find the power, or rate of energy consumption, \(P(t)=E^{\prime}(t)\) in units of megawatts (MW). b. At what time of day is the rate of energy consumption a maximum? What is the power at that time of day? c. At what time of day is the rate of energy consumption a minimum? What is the power at that time of day? d. Sketch a graph of the power function reflecting the times at which energy use is a minimum or maximum.

The population of a culture of cells after \(t\) days is approximated by the function \(P(t)=\frac{1600}{1+7 e^{-0.02 t}},\) for \(t \geq 0\). a. Graph the population function. b. What is the average growth rate during the first 10 days? c. Looking at the graph, when does the growth rate appear to be a maximum? d. Differentiate the population function to determine the growth rate function \(P^{\prime}(t)\). e. Graph the growth rate. When is it a maximum and what is the population at the time that the growth rate is a maximum?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.