The derivative of the function f(x) = (tan(πx/4))^4/πx can be found by using the chain rule and the quotient rule.

Step 1: Rewrite the function
Let's rewrite the function as f(x) = (tan(πx/4))^4 * (1/πx).

Step 2: Apply the chain rule to the first term
The chain rule states that if we have a function g(x) raised to a power n, the derivative of g(x)^n with respect to x is n * g(x)^(n-1) * g'(x), where g'(x) is the derivative of g(x).

In this case, the function g(x) is tan(πx/4), so applying the chain rule, the derivative of (tan(πx/4))^4 is 4 * (tan(πx/4))^3 * (sec^2(πx/4)) * (π/4).

Step 3: Apply the quotient rule to the second term
The quotient rule states that if we have a function h(x) divided by another function k(x), the derivative of h(x)/k(x) with respect to x is (h'(x) * k(x) - h(x) * k'(x)) / (k(x))^2, where h'(x) and k'(x) are the derivatives of h(x) and k(x), respectively.

In this case, the function h(x) is 1 and the function k(x) is πx, so applying the quotient rule, the derivative of (1/πx) is (0 * πx - 1 * π) / (πx)^2 = -1 / (πx)^2.

Step 4: Combine the derivatives
Now we have the derivatives of the two terms, so we can combine them. The derivative of f(x) = (tan(πx/4))^4 * (1/πx) is:

f'(x) = 4 * (tan(πx/4))^3 * (sec^2(πx/4)) * (π/4) - 1 / (πx)^2

Step 5: Simplify the expression
To simplify the expression, we can combine the terms and simplify the trigonometric function. The final derivative is:

f'(x) = (4π/16) * (tan(πx/4))^3 * (sec^2(πx/4)) - 1 / (πx)^2

Simplifying further:

f'(x) = (π/4) * (tan(πx/4))^3 * (sec^2(πx/4)) - 1 / (πx)^2

So, the derivative of the function f(x) = (tan(πx/4))^4/πx is (π/4) * (tan(πx/4))^3 * (sec^2(πx/4)) - 1 / (πx)^2.