Calculating nth Derivative of e^ax cos(bx+c)

In summary, the nth derivative of e^ax cos(bx+c) can be easily calculated using the given formula, but confusion may arise when using arctan to find theta, as tan is periodic with period π.
  • #1
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My question is about the nth derivative of e^ax cos(bx+c). Though i can calculate it easily but i am confused at one point.
When we calculate the first derivative we put a = r.cos(theta), b = r.sin(theta) (every thing is ok till here)
My confusion starts when we use (theta) = tan^-1(b/a) [tan inverse]
the reason for my confusion can be understood by:
suppose we have a = -1, b = 1
we put a = sqrt{2}*cos(3*pi/4)
b = sqrt{2}*sin(3*pi/4)
but the tan^-1(b/a) = tan^-1(-1) = -pi/4
but our theta is 3*pi/4
according to this theta our a will be 1 and b will be -1 which is different from our values of a and b
 
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  • #2
tan is periodic with period π, so arctan (-1) = -π/4 + kπ, for any integer k.
 

FAQ: Calculating nth Derivative of e^ax cos(bx+c)

1. What is the formula for calculating the nth derivative of e^ax cos(bx+c)?

The formula for calculating the nth derivative of e^ax cos(bx+c) is:
(-a)^n * e^ax * cos(bx+c-n*pi/2), where n is the order of the derivative.

2. How do you find the value of a and b in the formula for calculating the nth derivative?

The values of a and b can be found by comparing the given function with the general form of e^ax cos(bx+c). For example, if the given function is e^2x cos(3x+pi/4), then a=2 and b=3.

3. Can the nth derivative of e^ax cos(bx+c) be simplified further?

Yes, the nth derivative can be simplified to:
(-a)^n * e^ax * (cos(bx+c) * sin(n*pi/2) + sin(bx+c) * cos(n*pi/2)).
This simplification can be useful when evaluating the derivative at specific values of x.

4. How do you use the nth derivative of e^ax cos(bx+c) in practical applications?

The nth derivative of e^ax cos(bx+c) can be used to find the maximum and minimum values of the given function, as well as the points of inflection. It is also used in solving differential equations and in signal processing.

5. Is it possible to find the nth derivative of e^ax cos(bx+c) for non-integer values of n?

Yes, the formula for calculating the nth derivative can be extended to non-integer values of n using the concept of fractional calculus. However, the resulting expression may involve complex numbers, making it more challenging to interpret and use in practical applications.

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