Find derivative of complex sinusoidal function

In summary, the derivative of h(x) = 3e^{sin(x+2)} is h'(x) = 3e^{sin(x+2)}(cos(x+2)). The chain rule and derivative rules for the exponential function were used to determine this derivative.
  • #1
pbonnie
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Homework Statement


Find derivative of:
[itex] h(x) = 3e^{sin(x+2)} [/itex]


Homework Equations


chain rule of derivatives, product rule(?)


The Attempt at a Solution


I'm quite sure I'm doing this wrong. Because the exponent is a product, for the derivative of the exponent I would have to use the product rule? So:
[itex] h'(x) = sin(x+2)(3e^{sin(x+2)-1})(cos(x+2) + sin(1)) [/itex]

[itex] thank you for your help [/itex]
 
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  • #2
The exponent of 'e' is not a product, it is the sine function evaluated at (x+2). The derivative of the exponential function is not the same as x raised to a power.

Haven't you studied trig functions and the exponential function?
 
  • #3
This is me attempting to get back into math after 5 years, I'm quite rusty, I'm relearning everything so I forget sometimes.
Okay, so then the derivative of the exponent of e would be
[itex] cos(x+2) [/itex] ?
So it would be [itex] h′(x)=sin(x+2)(3e^{sin(x+2)−1})(cos(x+2))[/itex] ?
 
  • #4
oh.. or just
[itex] h'(x) = 3e^{sin(x+2)}(cos(x+2))[/itex]
Since it's of the form [itex] f(x) = a^{g(x)}[/itex]?
 
  • #5

1. What is a complex sinusoidal function?

A complex sinusoidal function is a mathematical function that involves a complex number, which is a number that contains both a real and imaginary part. It takes the form of f(x) = a + bi, where a and b are constants and i is the imaginary unit equal to the square root of -1.

2. How do you find the derivative of a complex sinusoidal function?

To find the derivative of a complex sinusoidal function, you can use the same rules as finding the derivative of a regular sinusoidal function. However, since complex numbers have both real and imaginary parts, you will need to apply the chain rule to the imaginary part as well. This involves taking the derivative of the imaginary part separately and multiplying it by the derivative of the real part.

3. What are the applications of complex sinusoidal functions?

Complex sinusoidal functions have various applications in fields such as physics, engineering, and signal processing. They are used to describe periodic phenomena, such as oscillations and waves, and are important in understanding the behavior of electrical circuits, mechanical systems, and electromagnetic radiation.

4. How do you graph a complex sinusoidal function?

To graph a complex sinusoidal function, you can plot the real and imaginary parts separately on a complex plane. The real part will be plotted on the x-axis, while the imaginary part will be plotted on the y-axis. This will result in a spiral-like curve, known as a helix, which represents the complex sinusoidal function.

5. Can complex sinusoidal functions have more than one frequency?

Yes, complex sinusoidal functions can have multiple frequencies. This is because the frequency of a complex sinusoidal function is determined by the coefficient of the imaginary unit i. If this coefficient is a complex number itself, it will result in multiple frequencies. This is often seen in applications involving harmonic oscillations, where the complex sinusoidal function can have both a fundamental frequency and its harmonics.

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