Differentiation refers to the process of finding the differential (or derivative) of a function. It represents the infinitesimal change in the output \( dy \) with respect to an infinitesimal change in input \( dx \).
Basic Differentiation Rules
\[ dy = y' \, dx \] — Total differential of a function
\[ d(Cu) = C \, du \] — Constant multiple rule
\[ d(u + v - w) = du + dv - dw \] — Sum and difference rule
\[ d(uv) = u \, dv + v \, du \] — Product rule
\[ d(uvw) = (vw) \, du + (uw) \, dv + (uv) \, dw \] — Product rule extension
\[ d\left( \frac{u}{v} \right) = \frac{v \, du - u \, dv}{v^2} \] — Quotient rule
\[ d(u^n) = n u^{n-1} \, du \] — Power rule
\[ d(\sin u) = \cos u \, du \]
\[ d(\cos u) = -\sin u \, du \]
Terminology
Differentiation: The process of computing a derivative or differential.
dy/dx: The rate of change of \( y \) with respect to \( x \).
du: A differential (infinitesimal change) in \( u \).
Chain rule: A rule to differentiate composite functions via their inner derivatives.
Infinitesimal: A quantity that is very small, approaching zero.
Applications
Used in physics to compute instantaneous change in position, velocity, and acceleration.
Essential in engineering for solving differential equations in systems and mechanics.
Applies to biology for modeling population dynamics and rates of reactions.
Used in economics for analyzing marginal cost, utility, and growth trends.
Forms the base of integral calculus through reverse operations.
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