Calculo De Derivadas May 2026

Introduction The derivative is one of the most powerful tools in calculus. At its core, it measures instantaneous change —the rate at which one quantity changes with respect to another. From predicting stock market trends to optimizing manufacturing costs and modeling the motion of planets, derivatives are indispensable in science, engineering, economics, and beyond.

This article provides a step-by-step guide to calculating derivatives, starting from the formal definition and progressing through essential rules, special techniques (implicit and logarithmic differentiation), and higher-order derivatives. For a function ( y = f(x) ), the derivative, denoted ( f'(x) ) or ( \fracdydx ), is defined as the limit of the difference quotient as the interval approaches zero:

[ f'(x) = \lim_h \to 0 \frac(x+h)^2 - x^2h = \lim_h \to 0 \fracx^2 + 2xh + h^2 - x^2h = \lim_h \to 0 (2x + h) = 2x ] calculo de derivadas

| Function | Derivative | |----------|------------| | ( x^n ) | ( n x^n-1 ) | | ( e^x ) | ( e^x ) | | ( a^x ) | ( a^x \ln a ) | | ( \ln x ) | ( \frac1x, x > 0 ) | | ( \log_a x ) | ( \frac1x \ln a ) | | ( \sin x ) | ( \cos x ) | | ( \cos x ) | ( -\sin x ) | | ( \tan x ) | ( \sec^2 x ) | | ( \cot x ) | ( -\csc^2 x ) | | ( \sec x ) | ( \sec x \tan x ) | | ( \csc x ) | ( -\csc x \cot x ) | | ( \arcsin x ) | ( \frac1\sqrt1-x^2 ) | | ( \arccos x ) | ( -\frac1\sqrt1-x^2 ) | | ( \arctan x ) | ( \frac11+x^2 ) | a. Implicit Differentiation Use when ( y ) is not isolated (e.g., ( x^2 + y^2 = 25 )). Differentiate both sides with respect to ( x ), treating ( y ) as a function of ( x ) and applying the chain rule whenever you differentiate ( y ).

While the limit definition is foundational, we rarely use it for complex functions. Instead, we rely on differentiation rules. a. Basic Rules | Rule | Formula | Example | |------|---------|---------| | Constant | ( \fracddx[c] = 0 ) | ( \fracddx[5] = 0 ) | | Power Rule | ( \fracddx[x^n] = n x^n-1 ) | ( \fracddx[x^4] = 4x^3 ) | | Constant Multiple | ( \fracddx[c \cdot f(x)] = c \cdot f'(x) ) | ( \fracddx[3x^2] = 6x ) | | Sum/Difference | ( (f \pm g)' = f' \pm g' ) | ( \fracddx[x^3 + x] = 3x^2 + 1 ) | b. Product Rule When two differentiable functions are multiplied: Introduction The derivative is one of the most

[ f'(x) = \lim_h \to 0 \fracf(x+h) - f(x)h ]

The slope of the tangent line to the curve at the point ( (x, f(x)) ). This article provides a step-by-step guide to calculating

[ \fracddx\left[\fracf(x)g(x)\right] = \fracf'(x) g(x) - f(x) g'(x)[g(x)]^2 ]

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