This is not a single function but rather an entire family of functions with the desired property. The problem above is a simple one. You have used other techniques to determine answers to more difficult questions. For example, you used integration by parts to find a family of functions that, when differentiated, yield . Other techniques of integration such as trigonometric substitutions, integration by partial fractions, and integration using tables of integrals, help with other classes of problems.
Differential equations is an extension of integral calculus.
For our purposes, it amounts to the development of techniques for finding functions that satisfy equations that involve derivatives of functions. The solution is simple to find. Other problems are more difficult to solve. That question is not simple to answer. We will find that the answer is no.