Read the derivative
A positive derivative means the function is locally increasing; a negative one means locally decreasing. Its magnitude measures steepness in the chosen units. A derivative of zero marks a stationary point, but that point could be a minimum, maximum, or neither. A sharp corner can make the derivative undefined.
In machine learning, derivatives tell how a loss changes when a parameter changes. Gradient descent steps against the gradient to reduce a differentiable objective. See partial derivatives and gradients and the chain rule.
Units and local meaning
A derivative has units of output per unit of input. If distance is measured in metres and time in seconds, ds/dt is measured in metres per second. It describes a local rate near the chosen point, not necessarily the average rate across a large interval. For f(x)=x², the average slope from x=3 to x=3+h is ((3+h)²−9)/h=6+h; as h approaches zero, the slope approaches 6.
A positive derivative means the function is increasing locally, and a negative one means decreasing. A zero derivative is only a candidate for a local extremum: f(x)=x³ has derivative zero at 0 but keeps increasing through it. A corner such as |x| has no single tangent slope at its cusp. Even when a derivative exists, a tangent-line approximation is reliable only near the point and for sufficiently smooth functions. In optimisation, derivatives describe local sensitivity; they do not tell whether a proposed step will reduce the objective unless the direction and step size are chosen appropriately.
Try this above
Move the point along the curve and compare the tangent slope at different locations. Check where the slope is zero and whether the curve actually has a maximum or minimum there.
A derivative can be estimated numerically with a finite difference, [f(x+h)−f(x)]/h, but choosing h is a trade-off. If h is too large, the secant slope differs from the tangent; if h is too small, rounding and cancellation can dominate. Automatic differentiation applies derivative rules through a computation graph and is usually more accurate than finite differences for machine-learning gradients. It still computes derivatives of the implemented function: bugs, discontinuities, or a poorly scaled loss remain problems.
For a position function s(t), the derivative ds/dt is instantaneous velocity and the second derivative is acceleration. This illustrates why the derivative's units depend on both quantities and why an average rate over an interval can differ from the instantaneous value. A derivative describes local behaviour; predicting far into the future requires a model for how the function itself evolves, not just its current slope.