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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?
- Tanh x
- Sech x
- Sinh x (correct)
- Csch 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?
- Coth x
- Sinh x
- Csch^2 x
- Sech^2 x (correct)
What is the derivative of y = Sinh^3 x?
What is the derivative of y = Sinh^3 x?
- 3 Sinh^2 x Cosh x (correct)
- 3 Cosh^2 x Sinh x
- Cosh^3 x
- Singh^2 x Cosh x
What is the derivative of y = ln(1 + tanh x)?
What is the derivative of y = ln(1 + tanh x)?
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'?
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)?
What is the derivative of y = Sin^-1(tan x)?
What is the derivative of y = Sin^-1(tan x)?
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?
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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