Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs.(a)    f(x) = xnf (n)(x) = (b)    f(x) = 1/xf (n)(x) =

(a) The function is f(x) = x^n.

The first derivative f'(x) = n*x^(n-1).

The second derivative f''(x) = n*(n-1)*x^(n-2).

The third derivative f'''(x) = n*(n-1)*(n-2)*x^(n-3).

From this pattern, we can see that the nth derivative f^(n)(x) = n! for n > 0 and f^(n)(x) = 0 for n > 1.

(b) The function is f(x) = 1/x = x^(-1).

The first derivative f'(x) = -1*x^(-2) = -1/x^2.

The second derivative f''(x) = 2*x^(-3) = 2/x^3.

The third derivative f'''(x) = -6*x^(-4) = -6/x^4.

From this pattern, we can see that the nth derivative f^(n)(x) = ((-1)^n)*n!/x^(n+1) for n > 0.