why ?
Since f^n(x) = -36 sin(6x), we know that n = 1.
We can determine that n = 1 because the function f^(n)(x) represents the nth derivative of the function f(x). In this case, f^n(x) represents the first derivative of f(x).
If we have f^n(x) = -36 sin(6x), this means that the first derivative of f(x) is equal to -36 sin(6x). Therefore, n must be 1, as it represents the first derivative.
To determine why n equals 1, we need to compare the given expression f^n(x) = -36 sin(6x) to the original expression f(x).
The expression f^n(x) represents the nth derivative of the function f(x). In this case, it means taking the derivative of f(x) n times.
Let's assume f(x) is an arbitrary function represented by f(x) = g(x).
To find the first derivative of f(x), we apply the derivative operator to g(x) and get f'(x) = g'(x).
If we apply the derivative operator to f'(x), we get f''(x) = g''(x).
Continuing this process, if we apply the derivative operator n times to f(x), we get f^n(x) = g^n(x).
Therefore, if f^n(x) = -36 sin(6x), it means that g^n(x) = -36 sin(6x).
Since the derivative of sin(6x) with respect to any variable is always sin(6x), we can conclude that g(x) must be -36.
In other words, f(x) = g(x) = -36.
Therefore, n must equal 1 because only one derivative is needed to obtain the function g(x) = -36 from f(x).