Derivatives and local change: the best nearby straight line

For a nonzero input step $h$, subtract the initial output from the final output and divide by the input change. This difference quotient is the slope of a secant line through two graph points. The derivative is its limit as $h$ tends to zero, if a finite common limit exists from allowed directions. A small step estimates the derivative; the limiting process defines it.

The units follow directly from this construction. A position in meters differentiated with respect to time in seconds gives meters per second. Differentiating velocity again gives acceleration in meters per second squared. A negative derivative means the output decreases when the input increases locally; it does not mean the output itself is negative. A vehicle can have positive position and negative velocity simultaneously.

For $f(x)=x^2$, expand $(a+h)^2$ before simplifying. Subtracting $a^2$ leaves $2ah+h^2$, and division by nonzero $h$ gives $2a+h$. The final limit is $2a$. This example exposes the role of algebra: it removes the apparently problematic denominator while preserving the quotient for every nearby nonzero step, just as with a removable hole.

The tangent slope therefore varies with the input even though the function has one fixed formula. At zero it is zero; at positive inputs it is positive; at negative inputs it is negative. A horizontal tangent is not the same as a constant function. The quadratic changes on either side of zero even though its first-order change vanishes exactly at that point.

A differentiable function is approximated near $a$ by its value there plus slope times input change. The approximation error, divided by the size of that change, tends to zero. For $f(x)=x^2$, the omitted term is exactly the square of the step. This gives a concrete reason to prefer smaller steps when using the tangent for prediction, provided numerical measurement error does not dominate.

Not every continuous function has a derivative. The absolute-value graph has a corner at zero: slopes approach minus one from the left and plus one from the right. A single tangent slope cannot represent both. In machine learning, derivatives guide optimization where the model is differentiable; nonsmooth points require additional conventions or tools. A derivative also describes sensitivity, not a causal effect unless the model supports that interpretation.