To find the derivative of the function y = tan(x/3), we use the chain rule.
The chain rule states: if a variable z depends on the variable y, which itself depends on the variable x, (so z depends on x by the way of y), then the derivative of z with respect to x can be found by first taking the derivative of z with respect to y and then multiplying it by the derivative of y with respect to x.
The derivative of tan(u) is sec^2(u). In this case, our "u" is x/3. Therefore, when we differentiate, we will have to apply the chain rule and multiply the derivative of tan(u) by the derivative of u, which in our case is 1/3.
Hence, if we let u = x/3, the derivative y' is (sec^2(u))*(1/3), or (sec^2(x/3))*(1/3).
Now, sec^2(u) = tan^2(u) + 1. Hence, y' = (tan^2(x/3)+1)*(1/3).
To evaluate the derivative at x = π/2, we substitute x = π/2 into y':
y'(π/2) = (tan^2(π/6) + 1) *(1/3) = 4/9.
So, the derivative of y = tan(x/3) is (tan^2(x/3)+1)*(1/3) and the derivative evaluated at x = π/2 is 4/9.
Therefore, in the given equation y=tan(x/3 ), y ′( π/2 ) = 4/9.