f(x)= e−(5x)⋅cos(3x)
Find the derivative
just plug in the product rule and the chain rule
f = e^(-5x)cos(3x)
f' = -5e^(-5x) cos(3x) + e^(-5x) (-3sin(3x))
= -e^(-5x) (5cos3x + 3sin3x)
The given function is f(x) = e^(-5x)⋅cos(3x). Let's break down the function and understand it step by step.
1. The function has two parts: e^(-5x) and cos(3x).
- e^(-5x) is the exponential function with the base e, where e is approximately 2.71828, and -5x is the exponent.
- cos(3x) is the cosine function, where 3x is the angle in radians.
2. To evaluate the function at a specific value of x, you simply substitute that value into the function.
- For example, if you want to evaluate f(2), substitute x = 2 into the function:
f(2) = e^(-5(2))⋅cos(3(2)) = e^(-10)⋅cos(6).
3. To simplify further, you can calculate the exponential and cosine separately.
- e^(-10) is equivalent to 1/e^10, which is approximately 0.0000454 (rounded to 7 decimal places).
- cos(6) is the cosine of 6 radians, which is approximately 0.96017 (rounded to 5 decimal places).
4. Finally, multiply the exponential and cosine values to get the result:
f(2) = (0.0000454)⋅(0.96017) = 0.00004363 (rounded to 8 decimal places).
Using the same steps, you can evaluate the function at any specific value of x. Just substitute the value into the function, calculate the exponential and cosine expressions separately, and multiply the results to find the final output.