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Questions and Answers
Given the function y = tan⁻¹(x) + cot⁻¹(x), what is the derivative dy/dx?
Given the function y = tan⁻¹(x) + cot⁻¹(x), what is the derivative dy/dx?
The derivative dy/dx is 0.
If x = a sin θ and y = a cos θ, using the chain rule, what is the expression for dy/dx?
If x = a sin θ and y = a cos θ, using the chain rule, what is the expression for dy/dx?
dy/dx = -tan(θ).
Prove that if y = tan⁻¹(x), the relationship (1 + x²) = (1 + y²) holds through implicit differentiation.
Prove that if y = tan⁻¹(x), the relationship (1 + x²) = (1 + y²) holds through implicit differentiation.
Differentiate both sides; confirming that dy/dx = 1/(1 + x²).
How do you differentiate the function f(x) = sin(x) + cos(x) using chain rule?
How do you differentiate the function f(x) = sin(x) + cos(x) using chain rule?
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What is the derivative of the function f(x) = arcsin(x) using the formula for derivatives of inverse trigonometric functions?
What is the derivative of the function f(x) = arcsin(x) using the formula for derivatives of inverse trigonometric functions?
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What is the derivative of the function $y = (x^4 + 5x^2 + 6)^{ rac{3}{2}}$ using the chain rule?
What is the derivative of the function $y = (x^4 + 5x^2 + 6)^{ rac{3}{2}}$ using the chain rule?
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Find the derivative of $y = (x+1)^{ rac{3}{4}}$ using the chain rule.
Find the derivative of $y = (x+1)^{ rac{3}{4}}$ using the chain rule.
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Calculate the derivative of $y = rac{x^3}{ ext{sqrt}(x^3 + 1) - 1}$ using implicit differentiation.
Calculate the derivative of $y = rac{x^3}{ ext{sqrt}(x^3 + 1) - 1}$ using implicit differentiation.
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Differentiate $ rac{x^3}{1+x^3}$ with respect to $x^3$. What is the result?
Differentiate $ rac{x^3}{1+x^3}$ with respect to $x^3$. What is the result?
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For the implicit function $y - xy - ext{sin}(y) = 0$, find $ rac{dy}{dx}$ where it exists.
For the implicit function $y - xy - ext{sin}(y) = 0$, find $ rac{dy}{dx}$ where it exists.
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How do you find the equation of the tangent line to the curve defined by $3x² - 7y² + 14y - 27 = 0$ at a specific point?
How do you find the equation of the tangent line to the curve defined by $3x² - 7y² + 14y - 27 = 0$ at a specific point?
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Differentiate the function $f(x) = (x + 2) \sin x$ with respect to $x$.
Differentiate the function $f(x) = (x + 2) \sin x$ with respect to $x$.
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When differentiating $y = \tan^{-1}(\sin 2x / (1 + \cos 2x))$, what rule do you apply and what is the derivative?
When differentiating $y = \tan^{-1}(\sin 2x / (1 + \cos 2x))$, what rule do you apply and what is the derivative?
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Explain how to prove that if $y = \tan (a \tan^{-1} x)$, then $(1 + x²) \frac{dy}{dx} - (1 + y²) \frac{dy}{dx} = 0$.
Explain how to prove that if $y = \tan (a \tan^{-1} x)$, then $(1 + x²) \frac{dy}{dx} - (1 + y²) \frac{dy}{dx} = 0$.
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Find $\frac{dy}{dx}$ if $y = \tan^7(2 \tan^{-1}x)$ and provide a brief justification.
Find $\frac{dy}{dx}$ if $y = \tan^7(2 \tan^{-1}x)$ and provide a brief justification.
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Study Notes
Differentiation Problems
- The document presents various problems related to differentiation, encompassing different techniques and concepts.
- The problems involve a diverse range of functions, including trigonometric functions, inverse trigonometric functions, and implicitly defined functions.
- The problems require the application of differentiation rules like the chain rule, the product rule, and the quotient rule.
- Differentiation techniques are used to find derivatives of functions and determine equations of tangent lines to curves.
- Trigonometric identities are utilized to simplify expressions and solve problems.
- Problems often involve finding the derivative (dy/dx) of functions with respect to x or other variables, such as t or y.
- Implicit differentiation is applied to find dy/dx when the function is defined implicitly in terms of x.
- Problems include proving and finding equations involving trigonometric functions and their inverses.
Example Problems
- Find dy/dx if y = tan(x) + cot⁻¹(x): This problem asks for the derivative of a combination of a trigonometric function and its inverse.
- If x = a sin θ and y = a cos θ, find dy/dx: This problem involves parametric equations and requires applying the chain rule for substitution.
- Problems involving y = tan⁻¹(x), and proving equations involving (1 + x²) and (1 + y²): These problems focus on the inverse tangent function and require applying differentiation and trigonometric identities.
- Problems involving functions including trigonometric functions (sin, cos, tan, cot): These problems involve differentiating trigonometric functions using the chain rule and other rules.
- Problems dealing with functions including inverse trigonometric functions (arcsin, arccos, arctan): These problems require applying differentiation techniques for inverse trigonometric functions.
General Structure
- The problems are arranged sequentially, each presented on its own line.
- The problems are likely part of a notebook or study material for a calculus course.
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Description
This quiz explores various differentiation problems, focusing on techniques such as the chain rule, product rule, and quotient rule. It covers a range of functions including trigonometric and inverse trigonometric functions, as well as implicit differentiation. Solve problems to find derivatives and equations of tangent lines.
