Calculus Formula Cheat Sheet for Limits, Derivatives

Calculus formula cheat sheet lists limits, derivatives, and integration formulas in an easy format, helping students understand change and motion and solve advanced mathematical problems step by step.

Calculus Formula Cheat Sheet

Topic	Formula	Description
Limit Definition	lim(x→a) f(x)	Value function approaches
Derivative Definition	f′(x) = lim(h→0) [f(x+h)−f(x)]/h	First principle
Power Rule	d/dx (xⁿ) = n xⁿ⁻¹	Basic derivative
Constant Rule	d/dx (c) = 0	Constant derivative
Constant Multiple	d/dx (cf) = c f′(x)	Scalar rule
Sum Rule	d/dx (f+g) = f′ + g′	Add functions
Product Rule	(fg)’ = f′g + fg′	Multiply functions
Quotient Rule	(f/g)’ = (f′g − fg′)/g²	Divide functions
Chain Rule	d/dx f(g(x)) = f′(g(x))g′(x)	Composite functions
eˣ Derivative	d/dx (eˣ) = eˣ	Exponential
aˣ Derivative	d/dx (aˣ) = aˣ ln a	General exponential
ln x Derivative	d/dx (ln x) = 1/x	Log derivative
logₐx Derivative	d/dx (logₐx) = 1/(x ln a)	Log base a
sin x	d/dx (sin x) = cos x	Trig derivative
cos x	d/dx (cos x) = −sin x	Trig derivative
tan x	d/dx (tan x) = sec²x	Trig derivative
Integral Definition	∫f(x)dx	Antiderivative
Power Rule (Int)	∫xⁿ dx = xⁿ⁺¹/(n+1) + C	Integration rule
∫1/x dx	ln	x
∫eˣ dx	eˣ + C	Exponential
∫aˣ dx	aˣ/ln a + C	General exponential
∫sin x dx	−cos x + C	Trig integral
∫cos x dx	sin x + C	Trig integral
∫sec²x dx	tan x + C	Trig integral
∫csc²x dx	−cot x + C	Trig integral
∫sec x tan x dx	sec x + C	Trig integral
∫csc x cot x dx	−csc x + C	Trig integral
Definite Integral	∫ₐᵇ f(x)dx = F(b)−F(a)	Area under curve
Substitution	∫f(g(x))g′(x)dx	u-substitution
Integration by Parts	∫u dv = uv − ∫v du	Product integration
Limit Important	lim(x→0) (sin x)/x = 1	Standard limit
Limit Important	lim(x→∞) (1 + 1/x)ˣ = e	Exponential limit
Slope Formula	f′(x)	Rate of change
Tangent Line	y − y₁ = m(x − x₁)	Tangent equation
Area Between Curves	∫(top − bottom)dx	Area difference
Volume (Disk)	π∫r² dx	Solid of revolution
Volume (Shell)	2π∫x f(x) dx	Cylindrical shell

Download PDF

Conclusion Calculus is a field of mathematics that deals in change and movement. It can be broadly split into two, that is, the differential calculus and integral calculus. Differential calculus is the study of change of quantities and the study of the accumulation and area is the domain of the integral calculus. Calculus is very much applicable in science, engineering, economics, and much more. It aids in learning such concepts as velocity, acceleration, growth, and decay. Calculus enables us to study complicated systems and forecast future behavior by means of limits, derivatives, and integrals. In physics, it is fundamental in the study of motion and forces, and in engineering, in the design of systems and structures. Calculus is also an important part of optimization problems, where we seek maximum or minimum values. Studying calculus enhances logical reasoning and critical thinking. It is a higher-level mathematical topic, based on algebra and trigonometry. Calculus is one of the tools that students employ to model the real world and resolve practical problems. In general, calculus is an effective mathematical device which makes the world understand how things vary and provides in finding the solutions to real-life issues. In summary, calculus helps us understand change and motion in a detailed way. It is widely used in science and technology and is essential for solving advanced real-world problems.