I assume this applies to linear functions (y=mx+c), but are there other functions that fit this description?
2026-05-05 21:36:02.1778016962
What type of functions have an average rate of change thats the same as the derivative?
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You are looking for differentiable function that satisfies
$$\frac{f(x)-f(x_{0})}{x-x_{0}}=f^{\prime}(x_{0})=\lim_{x\rightarrow x_{0}}\frac{f(x)-f(x_{0})}{x-x_{0}}\qquad (1)$$ for every $x_{0}$.
As user247327 commented, only linear functions satisfy (1) because, unlike nonlinear functions, they have the linear representation $f(x)=f(x_{0})+(x-x_{0})f^{\prime}((x_{0})$ at every point $x_{0}$.
Intuitively, linear functions are the only functions with the property that the slope of the tangent at every point equals the slope of every secant (chord) passing through that point.