/*! 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 39 Find the derivative of the follo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 39

Find the derivative of the following functions. $$y=\frac{1}{\sec z \csc z}$$

Short Answer

Expert verified
Question: Find the derivative of the function $$y = \frac{1}{\sec z \csc z}$$. Answer: The derivative of the function $$y = \frac{1}{\sec z \csc z}$$ is $$\frac{dy}{dz} = \cos^2 z - \sin^2 z$$.

Step by step solution

01

Rewrite the function in terms of sin and cos

$$y = \frac{1}{\sec z \csc z} = \frac{1}{\frac{1}{\cos z} \frac{1}{\sin z}} = \sin z \cos z$$
02

Apply the product rule and the chain rule to find the derivative

Since $$y = \sin z \cos z$$, its derivative is: $$\frac{dy}{dz} = (\sin z)' \cos z + \sin z (\cos z)'$$ Now we need to differentiate $$\sin z$$ and $$\cos z$$: $$\frac{d}{dz}(\sin z) = \cos z$$ $$\frac{d}{dz}(\cos z) = -\sin z$$ Plug these derivatives back into the expression: $$\frac{dy}{dz} = \cos z \cdot \cos z + \sin z (-\sin z) = \cos^2 z - \sin^2 z$$
03

Final Answer

The derivative of the function $$y = \frac{1}{\sec z \csc z}$$ is $$\boxed{\frac{dy}{dz} = \cos^2 z - \sin^2 z}$$.

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

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises \(63-68 .)\) b. Graph the tangent and normal lines on the given graph. $$\begin{aligned}&3 x^{3}+7 y^{3}=10 y\\\&\left(x_{0}, y_{0}\right)=(1,1)\end{aligned}$$

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

The Witch of Agnesi The graph of \(y=\frac{a^{3}}{x^{2}+a^{2}},\) where \(a\) is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi). a. Let \(a=3\) and find an equation of the line tangent to \(y=\frac{27}{x^{2}+9}\) at \(x=2\) b. Plot the function and the tangent line found in part (a).

Continuity of a piecewise function Let $$f(x)=\left\\{\begin{aligned} \frac{3 \sin x}{x} & \text { if } x \neq 0 \\ a\ \ \ \ \ & \text { if } x=0 \end{aligned}\right.$$ For what values of \(a\) is \(f\) continuous?

Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\) $$f(x)=x^{2}-2 x-3, \text { for } x \leq 1 ;(12,-3)$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.