Correcting Calculus Derivative: x(t) = A cos(ωt - \varphi)

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In summary, the derivative of x(t) is v(t) = -Aωsin(ωt + varphi). This is because the negative sign from the derivative of cosx is brought outside due to U substitution. However, there seems to be a typo on the Wikipedia page as the correct derivative is -Aωsin(ωt - varphi), not -Aωsin(ωt + varphi). This can be seen by evaluating at ω = 0 and varphi = π/2.
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Agrasin
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I'm brushing up on calculus. I don't see how this derivative works.

x(t) = A cos(ωt - [itex]\varphi[/itex])

v(t) = dx / dt = - Aωsin(ωt + [itex]\varphi[/itex])



I get that the derivative of cosx is -sinx. I get that the omega is brought outside the cos function because of U substitution.

Why does the the minus sign turn into a plus sign?

Btw I saw this on wikipedia: http://en.wikipedia.org/wiki/Simple_harmonic_motion
 
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Looks like a typo on the Wikipedia page. The correct derivative of ##A \cos(\omega t - \varphi)## is ##-A\omega\sin(\omega t - \varphi)##.

Since ##-\sin(x) = \sin(-x)##, an equivalent expression is ##A\omega\sin(-\omega t + \varphi)##.

But ##-A\omega\sin(\omega t + \varphi)## is not equal to either of these, as can easily be seen by evaluating at ##\omega = 0##, ##\varphi = \pi/2##.
 
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1. What is the purpose of correcting calculus derivative?

The purpose of correcting calculus derivative is to find the instantaneous rate of change of a variable with respect to another variable. This is useful in various fields of science and engineering, such as physics, economics, and engineering, to analyze and predict the behavior of systems.

2. How is the derivative of x(t) = A cos(ωt - ϕ) calculated?

The derivative of x(t) = A cos(ωt - ϕ) is calculated using the chain rule of differentiation. The derivative is found by multiplying the derivative of the outer function, cos(ωt - ϕ), with the derivative of the inner function, ωt - ϕ. This results in the derivative being -Aω sin(ωt - ϕ).

3. What do the variables A, ω, and ϕ represent in the equation?

A represents the amplitude of the function, which is the maximum displacement from the mean. ω represents the angular frequency, which determines the speed of the oscillations. ϕ represents the phase shift, which determines the horizontal shift of the function.

4. How does the value of A affect the derivative of x(t) = A cos(ωt - ϕ)?

The value of A does not affect the derivative of x(t) = A cos(ωt - ϕ). This is because the amplitude only affects the vertical scaling of the function, while the derivative only depends on the function's shape and not its magnitude.

5. Is it necessary to correct the calculus derivative for every function?

Yes, it is necessary to correct the calculus derivative for every function. This is because the derivative is affected by the shape of the function, and different functions will have different shapes and therefore require different correction methods.

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