Imagine you are given a set of data points $\{x_i,y_i\}$, supplemented by a list of known first derivatives $\{y'_i\}$.
How would you construct an interpolating function $y(x)$ (which satisfies $y(x_i)=y_i$ and $y'(x_i)=y'_i$), such that the derivative, calculated from this function has the smallest error.
The function is expected to be well behaved, consisting of piece-wise power functions, smoothely linked.
What would you do if you knew second derivatives as well and wanted to work with them instead of the first ones?
To settle this question: as already mentioned in the comments: one of the simplest approaches to this problem is (piecewise) Hermite interpolation (of which splines are a special case, since they have extra restrictions that guarantee continuity of their higher derivatives). In particular, there is an easy modification of the usual divided difference scheme to produce the unique polynomial that satisfies the function and derivative values supplied.
On the other hand, (polynomial) Hermite interpolation is not the be all and end all of interpolating given function values and derivatives. For instance, there are circumstances where a rational Hermite interpolant (a rational function that satisfies a given set of function and derivative values) might be more appropriate, when the function being approximated is expected to have poles or similar behavior, and if the function whose values are given is expected to be periodic, then certainly one needs to look into trigonometric Hermite interpolation. The computations needed for these variants are slightly more intricate than in the polynomial case, but if they give better results ("better" being up to your criteria of course), then the additional bother is certainly justifiable.