To find the 1000th derivative of the function f(x) = e^x * sin(x), we can follow these steps:
Step 1: Start by computing the derivative of f(x) with respect to x.
f'(x) = (e^x * sin(x))' = (e^x * sin(x)) + (e^x * sin(x))'
Step 2: Compute the derivative of sin(x) with respect to x using the product rule.
sin'(x) = cos(x)
Step 3: Apply the product rule to find the derivative of (e^x * sin(x)).
(e^x * sin(x))' = (e^x * cos(x)) + (e^x * sin(x))'
Step 4: Notice that we have a repeating pattern in each derivative. The derivative of (e^x * sin(x)) is always of the form:
(e^x * cos(x)) + (e^x * sin(x))'
Step 5: Based on the pattern identified, we can determine the general rule for the nth derivative of f(x).
n-th derivative of f(x) = (e^x * cos(x)) + (n-1) * (e^x * sin(x))'
Step 6: Now, let's find the second derivative (n = 2) to verify the general rule.
f''(x) = (e^x * cos(x)) + (2-1) * (e^x * sin(x))'
= (e^x * cos(x)) + (e^x * sin(x))'
Step 7: The second derivative matches our original function f'(x), confirming the general rule.
Step 8: Finally, using the general rule, we can find the 1000th derivative.
f^(1000)(x) = (e^x * cos(x)) + (1000-1) * (e^x * sin(x))'
Step 9: Simplifying the equation gives us the final solution for the 1000th derivative of f(x).
Therefore, the 1000th derivative of f(x) = (e^x * cos(x)) + 999(e^x * sin(x))'