Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Mathematics Problem: Differentiation**
**Problem Statement:**
Find the derivative \(\frac{dy}{dx}\) of the function \( y = \sec \left( 1 + \sqrt[3]{4 + \sin^{-1}(4 + \pi^2)} \right). \)
\[ \frac{dy}{dx} = \]
---
**Instructions for Solving the Problem:**
To find the derivative of this function, you will need to use the chain rule along with the derivatives of some trigonometric and inverse trigonometric functions. Here is a step-by-step approach:
1. **Identify the outer function and inner function:**
- The outer function here is the secant function (\(\sec\)).
- The inner function is \(1 + \sqrt[3]{4 + \sin^{-1}(4 + \pi^2)} \).
2. **Differentiate the outer function with respect to the inner function:**
- The derivative of \(\sec(u)\) with respect to \(u\) is \(\sec(u)\tan(u)\).
3. **Differentiate the inner function with respect to \(x\):**
- First, differentiate \(1 + \sqrt[3]{4 + \sin^{-1}(4 + \pi^2)}\) with respect to \(4 + \sin^{-1}(4 + \pi^2)\).
- Then, differentiate \(\sin^{-1}(4 + \pi^2)\) with respect to \(4 + \pi^2\).
4. **Combine the results using the chain rule:**
\[ \frac{dy}{dx} = \frac{d}{du} [\sec(u)] \cdot \frac{d}{dx} [1 + \sqrt[3]{4 + \sin^{-1}(4 + \pi^2)}] \]
By following these steps, you will be able to find the derivative of the given function.
*Note:* This problem features advanced calculus techniques including trigonometric and inverse trigonometric functions, and chain rule differentiation. Make sure you are familiar with these concepts to solve the problem correctly.
---
Add your solution in the provided space:
\[ \frac{dy}{dx} = \boxed{} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2d79032-32db-4435-bc8c-34ca2691b1b6%2F222eaef9-37da-4556-b920-1ea4742338fb%2F37vl3hh_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematics Problem: Differentiation**
**Problem Statement:**
Find the derivative \(\frac{dy}{dx}\) of the function \( y = \sec \left( 1 + \sqrt[3]{4 + \sin^{-1}(4 + \pi^2)} \right). \)
\[ \frac{dy}{dx} = \]
---
**Instructions for Solving the Problem:**
To find the derivative of this function, you will need to use the chain rule along with the derivatives of some trigonometric and inverse trigonometric functions. Here is a step-by-step approach:
1. **Identify the outer function and inner function:**
- The outer function here is the secant function (\(\sec\)).
- The inner function is \(1 + \sqrt[3]{4 + \sin^{-1}(4 + \pi^2)} \).
2. **Differentiate the outer function with respect to the inner function:**
- The derivative of \(\sec(u)\) with respect to \(u\) is \(\sec(u)\tan(u)\).
3. **Differentiate the inner function with respect to \(x\):**
- First, differentiate \(1 + \sqrt[3]{4 + \sin^{-1}(4 + \pi^2)}\) with respect to \(4 + \sin^{-1}(4 + \pi^2)\).
- Then, differentiate \(\sin^{-1}(4 + \pi^2)\) with respect to \(4 + \pi^2\).
4. **Combine the results using the chain rule:**
\[ \frac{dy}{dx} = \frac{d}{du} [\sec(u)] \cdot \frac{d}{dx} [1 + \sqrt[3]{4 + \sin^{-1}(4 + \pi^2)}] \]
By following these steps, you will be able to find the derivative of the given function.
*Note:* This problem features advanced calculus techniques including trigonometric and inverse trigonometric functions, and chain rule differentiation. Make sure you are familiar with these concepts to solve the problem correctly.
---
Add your solution in the provided space:
\[ \frac{dy}{dx} = \boxed{} \]
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