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Questions and Answers
What is the derivative of the function $y = 4sin x$?
What is the derivative of the function $y = 4sin x$?
- 4sin x
- 4
- 4tan x
- 4cos x (correct)
The derivative of $f(x) = ln(x^2 + 3)$ involves using the chain rule.
The derivative of $f(x) = ln(x^2 + 3)$ involves using the chain rule.
True (A)
What is the integral of $dx / (9 + x^2)$?
What is the integral of $dx / (9 + x^2)$?
arctan(x/3) + C
The derivative of $f(θ) = ln(sec(θ) + tan(θ))$ is ______.
The derivative of $f(θ) = ln(sec(θ) + tan(θ))$ is ______.
Match the following functions with their derivatives:
Match the following functions with their derivatives:
What is the result of the definite integral $\int_0^1 \sqrt{1 - x^2} , dx$?
What is the result of the definite integral $\int_0^1 \sqrt{1 - x^2} , dx$?
The integral $\int \frac{dx}{\sqrt{9 - x^2}}$ leads to an inverse trigonometric function.
The integral $\int \frac{dx}{\sqrt{9 - x^2}}$ leads to an inverse trigonometric function.
Find the derivative of $y = 4 - x$.
Find the derivative of $y = 4 - x$.
Flashcards
Derivative of y = 4x
Derivative of y = 4x
The derivative of a function y = 4x is simply 4. This is because the derivative of a constant times x is just the constant itself.
Derivative of f(x) = 4 - x
Derivative of f(x) = 4 - x
The derivative of the function f(x) = 4 - x is -1. The derivative of a constant is 0, and the derivative of -x is -1.
Derivative of y = 4sin(x)
Derivative of y = 4sin(x)
The derivative of y = 4sin(x) is 4cos(x). The derivative of sin(x) is cos(x), and the constant 4 stays the same.
Derivative of y = ln(2x)
Derivative of y = ln(2x)
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Derivative of ln(x^2 + 3)
Derivative of ln(x^2 + 3)
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Derivative of f(θ) = ln(sec(θ) + tan(θ))
Derivative of f(θ) = ln(sec(θ) + tan(θ))
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Derivative of y = ((2x+1)^10)/((x-3)^5*2)
Derivative of y = ((2x+1)^10)/((x-3)^5*2)
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Derivative of sin⁻¹(x²)
Derivative of sin⁻¹(x²)
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Study Notes
Calculus II - Chapter 7 Problem Set
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Finding Derivatives: The problem set includes exercises requiring the determination of derivatives for various functions.
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Functions: The document presents diverse functions, including constants, polynomials, trigonometric functions, natural logarithms, and composite functions.
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Derivatives of Simple Functions: Examples show differentiation of basic functions like y = 4, f(x) = 4 - x, and y = 4sinx.
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Chain Rule, Product Rule, Quotient Rule: The included exercises incorporate these differentiation rules, focusing on the application to more complex functions. Ex: finding dy/dx if y = (x² + 1)(x + 3)½ / (x - 1), for x > 1.
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Logarithmic Differentiation: Finding derivatives of functions involving natural logarithms, such as y = ln 2x and ln(x² + 3).
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Inverse Trigonometric Functions: The document contains a table outlining derivatives for inverse trigonometric functions (arcsin, arctan, arccos, arccot, arcsec, arccsc).
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Examples: Illustrative examples demonstrate finding the derivatives of composite functions involving inverse trigonometric functions and hyperbolic functions.
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Hyperbolic Functions: Derivatives and integrals related to hyperbolic functions (sinh, cosh, tanh, coth, sech, csch), are presented.
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Integral Evaluation: The document covers evaluating definite integrals related to inverse trigonometric functions.
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Examples of Definite and Indefinite Integrals: Various integral expressions are included, requiring the application of integral formulas for inverse trig, hyperbolic and related functions.
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Rate of Growth: Principles for comparing growth rates of different functions (e.g., exponential, polynomial) are introduced at the end.
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Description
This problem set focuses on finding derivatives of various functions, including constants, polynomials, and trigonometric functions. Exercises feature the application of differentiation rules such as the chain rule, product rule, and quotient rule, along with logarithmic and inverse trigonometric differentiation. It's designed to enhance understanding of derivative concepts in calculus.
