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The directional derivative is defined as an instantaneous rate of change, requiring an infinitesimal step rather than a finite one. By scaling the direction vector $\mathbf{v}$ with a scalar $h$ and taking the limit as $h \to 0$, the definition captures the local slope along that specific direction. Using the full vector $\mathbf{v}$ alone would represent a finite displacement, which fails to describe the derivative's nature as a limit of ratios over vanishingly small intervals.

Conditions: The function is differentiable at the point.; The direction is specified by a vector v.; h is a scalar approaching 0.

The directional derivative is defined as an instantaneous rate of change, requiring an infinitesimal step rather than a finite one. By scaling the direction vector $\mathbf{v}$ with a scalar $h$ and taking the limit as $h \to 0$, the definition captures the local slope along that specific direction. Using the full vector $\mathbf{v}$ alone would represent a finite displacement, which fails to describe the derivative's nature as a limit of ratios over vanishingly small intervals.

Conditions: The function is differentiable at the point.; The direction is specified by a vector v.; h is a scalar approaching 0.

Leibniz notation $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$ resembles algebraic cancellation, but derivatives are limits, not ordinary fractions. The rigorous justification relies on expressing increments exactly as $\Delta y = [g'(u)+\epsilon]\Delta u$ with $\epsilon \to 0$.

Conditions: Functions are differentiable.; The inner derivative $du/dx$ can be zero.; The argument requires rigorous limit definitions, not informal algebra.

Leibniz notation $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$ resembles algebraic cancellation, but derivatives are limits, not ordinary fractions. The rigorous justification relies on expressing increments exactly as $\Delta y = [g'(u)+\epsilon]\Delta u$ with $\epsilon \to 0$.

Conditions: Functions are differentiable.; The inner derivative $du/dx$ can be zero.; The argument requires rigorous limit definitions, not informal algebra.