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h'(x) = cosxcot(sin^3x)+cosxsinx ⠀ h(x) = sin(x) -2 (cos(t³) + t)dt

h'(x) = cosxcot(sin^3x)+cosxsinx ⠀ h(x) = sin(x) -2 (cos(t³) + t)dt

Calculus: Early Transcendentals
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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### Understanding Functions and Their Derivatives In calculus, it is important to understand how to work with functions and their derivatives. Below, we will discuss two important equations. #### 1. Integral Definition of a Function The function \( h(x) \) is defined as an integral: \[ h(x) = \int_{-2}^{\sin(x)} \left( \cos(t^3) + t \right) dt \] This integral represents the accumulation of the function \( \cos(t^3) + t \) over the interval from \(-2\) to \(\sin(x)\). #### 2. Derivative of the Function The derivative of the function \( h(x) \), represented as \( h'(x) \), is given by: \[ h'(x) = \cos x \cot(\sin^3 x) + \cos x \sin x \] This equation combines trigonometric functions such as cosine (\( \cos \)) and cotangent (\( \cot \)), showing how derivatives can involve complex combinations of functions. These equations illustrate crucial concepts in calculus, demonstrating the relationship between integrals, derivatives, and trigonometric functions. Understanding how to manipulate and differentiate these types of functions is fundamental in advanced mathematics and many applied fields.
Transcribed Image Text:### Understanding Functions and Their Derivatives In calculus, it is important to understand how to work with functions and their derivatives. Below, we will discuss two important equations. #### 1. Integral Definition of a Function The function \( h(x) \) is defined as an integral: \[ h(x) = \int_{-2}^{\sin(x)} \left( \cos(t^3) + t \right) dt \] This integral represents the accumulation of the function \( \cos(t^3) + t \) over the interval from \(-2\) to \(\sin(x)\). #### 2. Derivative of the Function The derivative of the function \( h(x) \), represented as \( h'(x) \), is given by: \[ h'(x) = \cos x \cot(\sin^3 x) + \cos x \sin x \] This equation combines trigonometric functions such as cosine (\( \cos \)) and cotangent (\( \cot \)), showing how derivatives can involve complex combinations of functions. These equations illustrate crucial concepts in calculus, demonstrating the relationship between integrals, derivatives, and trigonometric functions. Understanding how to manipulate and differentiate these types of functions is fundamental in advanced mathematics and many applied fields.
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