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The derivative at a point equals the slope of the tangent line there, representing instantaneous rate of change.
In calculus, the derivative of a function at a point (when it exists) measures how sensitively the function’s output changes with respect to a change in its input at that point. Geometrically, it equals the slope of the tangent line to the function’s graph at the chosen input value, giving the best linear approximation near that point. This is why the derivative is often described as the instantaneous rate of change: it is the limiting value of the ratio of an “instantaneous” change in the dependent variable to the corresponding change in the independent variable. Formally, for a function f(x), differentiability at a means the limit of the difference quotient exists: lim(h→0) [f(a+h)−f(a)]/h. This limit represents the slope approached by secant lines as the two points on the graph get arbitrarily close. Differentiation is the process of finding derivatives, and the derivative can be written using several notations, including Leibniz notation (dy/dx) and prime notation (f′(x)). Higher-order derivatives arise by repeatedly differentiating and have physical interpretations, such as velocity (first derivative with respect to time) and acceleration (second derivative with respect to time). The idea extends beyond single-variable functions. For functions of several variables, the derivative becomes a linear transformation that provides the best linear approximation near a point; the Jacobian matrix represents this transformation in coordinates, and its entries are partial derivatives. The derivative also connects to concepts like continuity and differentiability: if a function is differentiable at a point, it must be continuous there, though continuity alone does not guarantee differentiability.
The derivative at a point equals the slope of the tangent line there, representing instantaneous rate of change.
The derivative is defined as the limit of the difference quotient as the input change h approaches 0.
For multivariable functions, the derivative generalizes to a linear approximation represented by the Jacobian matrix (via partial derivatives).
The derivative of a function at a point is the limit of the difference quotient as the input increment approaches zero, giving the instantaneous rate of change.
The expression [f(a+h)−f(a)]/h that approximates the slope between two nearby points on the graph.
The slope of the line tangent to the graph at a point, which equals the derivative at that point when the derivative exists.
The derivative interpreted as the limiting ratio of an infinitesimal change in output to an infinitesimal change in input.
A matrix of partial derivatives that represents the linear transformation given by the derivative of a multivariable function at a point.
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