Derivatives formulas

All 22 formulas in this topic

Definition of the Derivativef′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}The Definition of the Derivative finds the instantaneous rate of change at a point using limits; the foundation for all derivative calculations.Power Ruleddxxn=nxn−1\frac{d}{dx}x^{n} = nx^{n-1}The Power Rule states that the derivative of x^n is nx^{n-1}, making it the essential tool for differentiating any polynomial expression.Product Rule(fg)′=f′g+fg′(fg)' = f'g + fg'Product Rule lets you find the derivative of a product of two functions without multiplying them out first; use it when the product is complex.Quotient Rule(fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}The Quotient Rule gives you the derivative of a fraction by combining the derivatives and original functions of the numerator and denominator.Chain Ruleddxf(g(x))=f′(g(x)) g′(x)\frac{d}{dx}f\big(g(x)\big) = f'\big(g(x)\big)\,g'(x)The Chain Rule tells you how to differentiate a composite function—when one function is plugged into another—by multiplying the outer and inner derivatives.Derivative of Sin xddxsin⁡x=cos⁡x\frac{d}{dx}\sin x = \cos xThe derivative of sin x is cos x, giving the slope of the sine curve; use this rule when differentiating any sine function.Derivative of Cos xddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x = -\sin xThe derivative of cosine x equals negative sine x, the fundamental rule for finding rates of change of oscillating cosine functions.Derivative of Tan xddxtan⁡x=sec⁡2x\frac{d}{dx}\tan x = \sec^2 xThe derivative of tan x equals sec squared x, needed to find slopes of tangent curves and solve optimization and related-rates problems.Derivative of e^xddxex=ex\frac{d}{dx}e^{x} = e^{x}The derivative of e^x is e^x itself—the unique exponential function used to model continuous growth, radioactive decay, and other rates of change.Derivative of ln xddxln⁡x=1x\frac{d}{dx}\ln x = \frac{1}{x}The derivative of ln x is 1/x, giving you the rate of change of the natural logarithm — essential for calculus problems involving logarithmic functions.Derivative of a^xddxax=axln⁡a\frac{d}{dx}a^{x} = a^{x}\ln aFind the rate of change of exponential functions with any base a using the formula a^x times ln(a), essential for modeling growth.Implicit Differentiationddx[F(x,y)]=0  ⟹  dydx\frac{d}{dx}\big[F(x,y)\big] = 0 \implies \frac{dy}{dx}Use implicit differentiation to find dy/dx when an equation defines y implicitly instead of as an explicit function of x.Logarithmic Differentiationy=f(x)g(x)  ⟹  ln⁡y=g(x)ln⁡f(x)y = f(x)^{g(x)} \implies \ln y = g(x)\ln f(x)Solve derivatives of functions like x^x where both base and exponent depend on x using logarithms and implicit differentiation.Higher Order Derivativesf′′(x)=ddxf′(x)f''(x) = \frac{d}{dx}f'(x)Higher order derivatives describe how the rate of change is changing; use them to find concavity and acceleration in functions.Critical Pointsf′(c)=0 or f′(c) undefinedf'(c) = 0 \text{ or } f'(c) \text{ undefined}Critical points mark where a function's derivative is zero or undefined, revealing local extrema and where the graph changes direction.First Derivative Testf′ changes +→−  ⟹  local max at cf' \text{ changes } + \to - \implies \text{local max at } cClassify critical points as local maxima or minima by checking whether the derivative changes sign before and after the point.Second Derivative Testf′(c)=0,;f′′(c)>0  ⟹  local minf'(c) = 0, ; f''(c) > 0 \implies \text{local min}The Second Derivative Test determines whether a critical point is a local minimum or maximum using the concavity at that point.Concavity and Inflection Pointsf′′(x)=0 and sign changesf''(x) = 0 \text{ and sign changes}Pinpoint where a function's curve changes from bending upward to downward, revealing the transition point between concave up and concave down.Mean Value Theoremf′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}The Mean Value Theorem says there exists at least one point in an interval where a function's instantaneous rate equals its average rate.Related RatesdVdt=dVdr⋅drdt\frac{dV}{dt} = \frac{dV}{dr}\cdot\frac{dr}{dt}Related Rates connects the rate of change of one quantity to the rate of change of another quantity through implicit differentiation.Linear ApproximationL(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x - a)Use the tangent line at a known point to estimate a function's value nearby when exact calculations are difficult or unnecessary.Optimizationmaximise f subject to a constraint g=0\text{maximise } f \text{ subject to a constraint } g = 0Find the maximum or minimum value of a function subject to a constraint: the fundamental tool for real-world decision-making problems.

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