Remember, since you want to find the derivative of f(x) at x = 1, you must differentiate before substituting in your x = 1.
f(x) = tan(x)/(1+x)
Let u(x) = tan(x) and v(x) = (1+x)
Using the product rule, f'(x) = ( u'(x)v(x)-u(x)v'(x) ) / v(x)^2
u'(x) = sec(x) = 1/cos(x)
v'(v) = 1
Therefore, f'(x) = ( (1/cos(x))(1+x)-tan(x)(1) ) / (1+x)^2
f'(x) = ( (1/cos(x))(1+x)-tan(x) ) / (1+x)^2
f'(0) = ( (1/cos(0))(1+0)-tan(0) ) / (1+0)^2
f'(0) = ( (1/1)(1)-0 ) / (1)^2 = 1/1 = 1
Hope that helps.
