The yield curve and smile curve (or surface) are the output of curve fitting engines, calibrated using a lot of commercial market data, discarding outliers. These are the most valuable soft market data objects. Directly used by decision makers, including trade pricing. Each trading desk guards these curves as highly proprietary trading secrets.
However, these curves are imprecise. You shouldn’t compute the gradient at each point on these curves. The most you do with such gradient is computing the gradient at the anchor vol point, which gives you the skew of the entire curve.
In contrast, the price-vs-impliedYield curve and the valuation-vs-spot curve are a different class of curves. (The valuation-vs-impliedVol curve is another example.) I call these “range of possibilities” curves. They are mathematically precise enough to let us compute gradient at every point. You get duration and delta. These are known as sensitivities, essential soft market data for risk management.