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Questions and Answers
What is the derivative of the function y = Cosh x?
What is the derivative of the function y = Cosh x?
Which of the following correctly represents the derivative of y = tanh x?
Which of the following correctly represents the derivative of y = tanh x?
What is the derivative of y = Sinh^3 x?
What is the derivative of y = Sinh^3 x?
What is the derivative of y = ln(1 + tanh x)?
What is the derivative of y = ln(1 + tanh x)?
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For the function y = √(Sin x + Sinh x), what is the derivative y'?
For the function y = √(Sin x + Sinh x), what is the derivative y'?
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Which of the following is the derivative of y = Cot^-1(ln x)?
Which of the following is the derivative of y = Cot^-1(ln x)?
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What is the derivative of y = Sin^-1(tan x)?
What is the derivative of y = Sin^-1(tan x)?
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In the context of implicit differentiation, what does the equation y * g'(x, y) = -x * g'(x, y) imply?
In the context of implicit differentiation, what does the equation y * g'(x, y) = -x * g'(x, y) imply?
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What is the derivative of the function y = Sec^-1(Sin x)?
What is the derivative of the function y = Sec^-1(Sin x)?
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Study Notes
Hyperbolic Functions
- Derivatives of Hyperbolic Functions:
- y = Sinh x : y' = Cosh x
- y = Cosh x : y' = Sinh x
- y = tanh x : y'= Sech x
- y = Sech x : y' = -Sech x tanh x
- y = Csch x : y' = -Csch x Coth x
- y = Coth x : y' = -Csch^2 x
Generalization of Hyperbolic Function Derivative Rules
- y = Sinh f(x)
- y' = f' (x) Cosh f(x)
Inverse Hyperbolic Functions
- y = Sin^-1 x
- y' = 1/(√(1 - x^2))
Implicit Differentiation
- Implicit Differentiation: A method for finding the derivative of a function defined implicitly. In this case, y is not explicitly defined as a function of x.
- General form: g(x, y) = 0
- Formula: y * g'(x, y) = -x * g'(x, y)
- Provides the derivative of y with respect to x.
Parametric Differentiation
- Parametric Differentiation: A method for finding the derivative of a function defined parametrically. In this case, x and y are expressed as functions of a parameter t.
- General form: x = y(t) , y = g(t)
- Formula: y' = g'(t) / y'(t)
- Provides the derivative of y with respect to x.
Logarithmic Differentiation
- Logarithmic Differentiation: A method for finding the derivative of a function that involves simplifying calculations by taking the natural logarithm of both sides of the equation.
- General form: y = f(x)
- Formula: y' = (f'(x)) / (f(x))
- This method is useful in situations where direct differentiation is complex.
Normal Differentiation
- y = P(x)
- y' = P'(x)
Implicit Differentiation
- Implicit Differentiation: A method for finding the derivative of a function defined implicitly. In this case, y is not explicitly defined as a function of x.
- General form: g(x, y) = 0
- Formula: y * g'(x, y) = -x * g'(x, y)
- Provides the derivative of y with respect to x.
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Description
This quiz covers the derivatives of hyperbolic functions, the rules for generalization, and the concepts of inverse hyperbolic functions. Additionally, it explores implicit and parametric differentiation methods. Test your understanding of these key calculus topics!
