What is the derivative with respect to x of the function f(x)=1+x^3+ln(x), x>0 ?

Despite the hideous view, we can apply to this function the same methodology as all the other ones: break it down to pieces. What we mean is that we recognize three terms inside f(x): one is the number 1 alone, a contant with derivative 0; another is the plolynomial function x^3 with derivative 3x^2; and the last one is the natural logarithm (in base e) with derivative 1/x. Note that the last term makes sense because we don't divide by 0 since our domain x>0 excludes that possibility. Finally, the derivative is lineal, meaning that the derivative of the sum is the sum of the derivative. This allows us to write the derivative of f with respect to x: df/dx(x)= 3x^2+1/x.