/*! 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 56 Find the derivative of the funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 56

Find the derivative of the function. \(y=25 \arcsin \frac{x}{5}-x \sqrt{25-x^{2}}\)

Short Answer

Expert verified
The derivative of the function \(y=25 \arcsin \frac{x}{5}-x \sqrt{25-x^{2}}\) is \(y'(x) = 5\frac{1}{\sqrt{25 - x^2}} - \sqrt{25 - x^2} + \frac{x^2}{\sqrt{25 - x^2}}\).

Step by step solution

01

Identify the parts of the function

Split the function into two parts as follows: \(f(x) = 25 \arcsin(\frac{x}{5})\) and \(g(x) = -x\sqrt{25 - x^2}\). These parts will be differentiated separately.
02

Find the derivative of the first part - \(f(x)\)

The derivative of \(f(x) = 25 \arcsin(\frac{x}{5})\), using the chain rule, is \(f'(x) = 25\frac{1}{\sqrt{1 - \left(\frac{x}{5}\right)^2}} \cdot \frac{1}{5} = 5\frac{1}{\sqrt{25 - x^2}}\).
03

Find the derivative of the second part - \(g(x)\)

The derivative of \(g(x)=-x \sqrt{25-x^2}\), using the product rule and chain rule, is \(g'(x) = -\sqrt{25 - x^2} - x\left(\frac{1}{2\sqrt{25 - x^2}} \cdot -2x\right) = -\sqrt{25 - x^2} + \frac{x^2}{\sqrt{25 - x^2}}\).
04

Combine the derivatives

The derivative of the total function \(y(x) = f(x) + g(x)\) is given by the sum of the derivatives of \(f(x)\) and \(g(x)\), i.e., \(y'(x) = f'(x) + g'(x) = 5\frac{1}{\sqrt{25 - x^2}} - \sqrt{25 - x^2} + \frac{x^2}{\sqrt{25 - x^2}}\) .

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!

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

Horizontal Motion The position function of a particle moving along the \(x\) -axis is \(x(t)=A e^{k t}+B e^{-k t},\) where \(A, B,\) and \(k\) are positive constants. (a) During what times \(t\) is the particle closest to the origin? (b) Show that the acceleration of the particle is proportional to the position of the particle. What is the constant of proportionality?

(a) Prove that arctan \(x+\arctan y=\arctan \frac{x+y}{1-x y}, x y \neq 1\) (b) Use the formula in part (a) to show that \(\quad \arctan \frac{1}{2}+\arctan \frac{1}{3}=\frac{\pi}{4}\).

Find the derivative of the function. \(f(x)=\arctan e^{x}\)

Modeling Data A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate \(R\) (in liters per hour) at time \(t\) (in hours) is given in the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} \\ \hline R & {425} & {240} & {118} & {71} & {36} \\ \hline\end{array} $$ (a) Use the regression capabilities of a graphing utility to find a linear model for the points \((t, \ln R) .\) Write the resulting equation of the form \(\ln R=a t+b\) in exponential form. (b) Use a graphing utility to plot the data and graph the exponential model. (c) Use the definite integral to approximate the number of liters of chemical released during the 4 hours.

Prove or disprove: there is at least one straight line normal to the graph of \(y=\cosh x\) at a point \((a, \cosh a)\) and also normal to the graph of \(y=\sinh x\) at a point \((c, \sinh c)\) [At a point on a graph, the normal line is the perpendicular to the tangent at that point. Also, cosh \(x=\left(e^{x}+e^{-x}\right) / 2\) and \(\sinh x=\left(e^{x}-e^{-x}\right) / 2 . ]\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.