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2. QUICK DERIVATIVES AND INTEGRALS a) y = 13x+ + 3e 4x5 Find y'. %3D

2. QUICK DERIVATIVES AND INTEGRALS a) y = 13x+ + 3e 4x5 Find y'. %3D

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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I really need help figuring out how to do this problem, using the cheat sheet formula is needed in this equation
**2. QUICK DERIVATIVES AND INTEGRALS** a) \( y = 13x^4 + \frac{2}{3e^{4x^5}} \) Find \( y' \). --- This section introduces a calculus problem focused on finding the derivative of a function. The given function \( y \) is composed of a polynomial and an exponential component. The task requires applying derivative rules for both types of functions.
Transcribed Image Text:**2. QUICK DERIVATIVES AND INTEGRALS** a) \( y = 13x^4 + \frac{2}{3e^{4x^5}} \) Find \( y' \). --- This section introduces a calculus problem focused on finding the derivative of a function. The given function \( y \) is composed of a polynomial and an exponential component. The task requires applying derivative rules for both types of functions.
# Calculus Cheat Sheet ### The Basics - **Function Basics**: - \( x \rightarrow f(x) \rightarrow y \) (where \( y \) can be \( +, -, 0 \)) - \( x \rightarrow f'(x) \) for slope, which can be \( +, -, 0 \). - Maxima, minima, and horizontal points of inflection (H.P.I.). - \( x \rightarrow f''(x) \) for concavity; \( \cap \), \( \cup \), and \( \Phi \) for point of inflection. ### Derivatives - \( y = x^n \rightarrow y' = nx^{n-1} \) - \( y = u^n \rightarrow y' = nu^{n-1}u' \) - \( y = e^u \rightarrow y' = u'e^u \) - \( y = \ln u \rightarrow y' = \frac{u'}{u} \) \( u \) is a function, \( x \) is a variable, \( e \) and \( n \) are constants. ### Products and Quotients - Product Rule: - \( y = uv \rightarrow y' = u'v + uv' \) - Quotient Rule: - \( y = \frac{u}{v} \rightarrow y' = \frac{u'v - uv'}{v^2} \) \( u \) and \( v \) are functions. ### Integrals - Power Rule: - \( y = \int x^n dx \rightarrow \frac{x^{n+1}}{n+1} + K \), \( n \neq -1 \) - Integrals involving \( u \) substitution and exponential functions. - Steps for Integration: 1. Simplify the integral. 2. Identify \( u \), find \( u' \). 3. Set what is known and what is needed. 4. Align with known templates. 5. Perform the integral. ### Logs and Exponents - Exponential Functions: - \( y = a^u \rightarrow y' = a^u \ln a \) - \( y = a^x \rightarrow y' = a^x \ln a \) - Logarithmic Functions
Transcribed Image Text:# Calculus Cheat Sheet ### The Basics - **Function Basics**: - \( x \rightarrow f(x) \rightarrow y \) (where \( y \) can be \( +, -, 0 \)) - \( x \rightarrow f'(x) \) for slope, which can be \( +, -, 0 \). - Maxima, minima, and horizontal points of inflection (H.P.I.). - \( x \rightarrow f''(x) \) for concavity; \( \cap \), \( \cup \), and \( \Phi \) for point of inflection. ### Derivatives - \( y = x^n \rightarrow y' = nx^{n-1} \) - \( y = u^n \rightarrow y' = nu^{n-1}u' \) - \( y = e^u \rightarrow y' = u'e^u \) - \( y = \ln u \rightarrow y' = \frac{u'}{u} \) \( u \) is a function, \( x \) is a variable, \( e \) and \( n \) are constants. ### Products and Quotients - Product Rule: - \( y = uv \rightarrow y' = u'v + uv' \) - Quotient Rule: - \( y = \frac{u}{v} \rightarrow y' = \frac{u'v - uv'}{v^2} \) \( u \) and \( v \) are functions. ### Integrals - Power Rule: - \( y = \int x^n dx \rightarrow \frac{x^{n+1}}{n+1} + K \), \( n \neq -1 \) - Integrals involving \( u \) substitution and exponential functions. - Steps for Integration: 1. Simplify the integral. 2. Identify \( u \), find \( u' \). 3. Set what is known and what is needed. 4. Align with known templates. 5. Perform the integral. ### Logs and Exponents - Exponential Functions: - \( y = a^u \rightarrow y' = a^u \ln a \) - \( y = a^x \rightarrow y' = a^x \ln a \) - Logarithmic Functions
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