Science:Math Exam Resources/Courses/MATH110/April 2019/Question 01 (g)/Solution 1

The function $2\ln x$ has derivative ${\frac {2}{x}}$. The function ${\frac {x^{5}}{20}}$ has derivative ${\frac {x^{4}}{4}}$. Constants have derivative 0. Therefore any function of the form

$f(x)=2\ln x+{\frac {x^{4}}{4}}+C$,

where $C$ is any constant, will have the correct derivative. In order to see this, notice that:

${\frac {d}{dx}}[2\ln x+{\frac {x^{5}}{20}}+C]={\frac {d}{dx}}[2\ln x]+{\frac {d}{dx}}[{\frac {x^{5}}{20}}]+{\frac {d}{dx}}[C]={\frac {2}{x}}+{\frac {x^{4}}{4}}$.