Calculus Reference
Every derivative and integral you need in one organized table — power, exponential, logarithmic, trigonometric, inverse-trig and hyperbolic rules — with one-tap links to our interactive derivative and integral solvers so you can work your own problem right away.
Common derivatives
Power & Roots
- d/dx(c)=0
- d/dx(x)=1
- d/dx(xⁿ)=n·xⁿ⁻¹
- d/dx(√x)=1/(2√x)
Exponential & Log
- d/dx(eˣ)=eˣ
- d/dx(aˣ)=aˣ·ln a
- d/dx(ln x)=1/x
- d/dx(logₐx)=1/(x·ln a)
Trigonometric
- d/dx(sin x)=cos x
- d/dx(cos x)=−sin x
- d/dx(tan x)=sec²x
- d/dx(cot x)=−csc²x
- d/dx(sec x)=sec x·tan x
- d/dx(csc x)=−csc x·cot x
Inverse Trig
- d/dx(arcsin x)=1/√(1−x²)
- d/dx(arccos x)=−1/√(1−x²)
- d/dx(arctan x)=1/(1+x²)
Rules
- (f ± g)′=f′ ± g′
- (c·f)′=c·f′
- (f·g)′=f′g + fg′
- (f/g)′=(f′g − fg′)/g²
- (f(g(x)))′=f′(g(x))·g′(x)
Hyperbolic
- d/dx(sinh x)=cosh x
- d/dx(cosh x)=sinh x
- d/dx(tanh x)=sech²x
These are the standard elementary results. Every integral can be checked by differentiating the answer to recover the original function — for example d/dx(−cos x) = sin x confirms ∫sin x dx = −cos x + C.
Common derivatives
Power & Roots
- d/dx(c) = 0
- d/dx(x) = 1
- d/dx(xⁿ) = n·xⁿ⁻¹
- d/dx(√x) = 1/(2√x)
Exponential & Log
- d/dx(eˣ) = eˣ
- d/dx(aˣ) = aˣ·ln a
- d/dx(ln x) = 1/x
- d/dx(logₐx) = 1/(x·ln a)
Trigonometric
- d/dx(sin x) = cos x
- d/dx(cos x) = −sin x
- d/dx(tan x) = sec²x
- d/dx(cot x) = −csc²x
- d/dx(sec x) = sec x·tan x
- d/dx(csc x) = −csc x·cot x
Inverse Trig
- d/dx(arcsin x) = 1/√(1−x²)
- d/dx(arccos x) = −1/√(1−x²)
- d/dx(arctan x) = 1/(1+x²)
Rules
- (f ± g)′ = f′ ± g′
- (c·f)′ = c·f′
- (f·g)′ = f′g + fg′
- (f/g)′ = (f′g − fg′)/g²
- (f(g(x)))′ = f′(g(x))·g′(x)
Hyperbolic
- d/dx(sinh x) = cosh x
- d/dx(cosh x) = sinh x
- d/dx(tanh x) = sech²x
Common integrals
Power & Roots
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
- ∫1/x dx = ln|x| + C
- ∫1 dx = x + C
Exponential & Log
- ∫eˣ dx = eˣ + C
- ∫aˣ dx = aˣ/ln a + C
- ∫ln x dx = x·ln x − x + C
Trigonometric
- ∫sin x dx = −cos x + C
- ∫cos x dx = sin x + C
- ∫sec²x dx = tan x + C
- ∫csc²x dx = −cot x + C
- ∫sec x·tan x dx = sec x + C
- ∫tan x dx = −ln|cos x| + C
- ∫cot x dx = ln|sin x| + C
Inverse Trig
- ∫1/(1+x²) dx = arctan x + C
- ∫1/√(1−x²) dx = arcsin x + C
- ∫1/(a²+x²) dx = (1/a)·arctan(x/a) + C
Rules
- ∫(f ± g) dx = ∫f dx ± ∫g dx
- ∫c·f dx = c·∫f dx
- ∫u dv = u·v − ∫v du
About the Calculus Reference
This is the calculus cheat sheet students reach for most: the common derivatives and indefinite integrals, organized exactly the way they are taught. Switch between the Derivatives and Integrals tabs and browse the cards by type — power and root rules (including d/dx(xⁿ) = n·xⁿ⁻¹ and ∫xⁿ dx = xⁿ⁺¹/(n+1) + C), exponential and logarithmic rules, the six trigonometric functions and their inverses, the differentiation and integration rules (product, quotient, chain, linearity and integration by parts), and the basic hyperbolic derivatives. Every formula is written in clean Unicode notation — no images — so it copies cleanly into your notes and is easy for assistive technology and AI to read. What sets this page apart from a static PDF is that it is wired into UnitConv's interactive calculators: when you find the rule you need, tap "Solve your own derivative" or "Solve your own integral" to open our step-by-step solvers and run your actual problem. Use it as a homework reference, an exam revision list, or a quick lookup while you are working through a calculus set.
How to use this reference
- 1 Pick the Derivatives or Integrals tab depending on what you are looking up.
- 2 Find the rule in the grouped cards — power, exponential, log, trig, inverse trig, rules or hyperbolic — and copy it into your work.
- 3 Tap "Solve your own derivative" or "Solve your own integral" to open the interactive calculator and run your own function.
How derivatives and integrals connect
Differentiation and integration are inverse operations, which is the whole point of the Fundamental Theorem of Calculus: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. That is why every entry in the integral table can be verified by differentiating the answer: • ∫xⁿ dx = xⁿ⁺¹/(n+1) + C because d/dx(xⁿ⁺¹/(n+1)) = (n+1)·xⁿ/(n+1) = xⁿ (valid for n ≠ −1; the missing case ∫1/x dx = ln|x| + C fills the gap). • ∫sin x dx = −cos x + C because d/dx(−cos x) = sin x, and ∫cos x dx = sin x + C because d/dx(sin x) = cos x. • ∫1/(1+x²) dx = arctan x + C because d/dx(arctan x) = 1/(1+x²). The indefinite integral carries a constant of integration "+ C" because differentiating any constant gives zero, so the antiderivative is only determined up to that constant. The differentiation rules (product, quotient and chain) and their integral counterparts (linearity and integration by parts, ∫u dv = u·v − ∫v du, which is just the product rule run backwards) let you extend these building blocks to almost any elementary function.
Frequently asked questions
What are the most common derivatives to memorize?
The essentials are the power rule d/dx(xⁿ) = n·xⁿ⁻¹, the exponential and log rules d/dx(eˣ) = eˣ and d/dx(ln x) = 1/x, and the trig derivatives d/dx(sin x) = cos x, d/dx(cos x) = −sin x and d/dx(tan x) = sec²x. Combined with the product, quotient and chain rules, these let you differentiate almost any elementary function you meet in a first calculus course.
Why does every indefinite integral have a "+ C"?
Because differentiating a constant gives zero, any two antiderivatives of the same function differ only by a constant. So the indefinite integral is not a single function but a whole family, written with the constant of integration "+ C". For example ∫cos x dx = sin x + C, and both sin x + 3 and sin x − 7 are valid antiderivatives. The constant is pinned down only when you have an initial condition or evaluate a definite integral.
How do I check that an integral is correct?
Differentiate your answer — if you get back the original integrand, the integral is right. This works because differentiation and integration are inverse operations. For instance, to confirm ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, differentiate the right side: d/dx(xⁿ⁺¹/(n+1)) = (n+1)·xⁿ/(n+1) = xⁿ, which is exactly the integrand. This self-check is the fastest way to catch a mistake.
What is integration by parts?
Integration by parts is the integral counterpart of the product rule: ∫u dv = u·v − ∫v du. You use it when the integrand is a product of two different kinds of function — for example x·eˣ or x·ln x — choosing u and dv so that the remaining integral ∫v du is simpler than the original. A common guide for picking u is the LIATE order (logarithmic, inverse trig, algebraic, trig, exponential).
Related tools
Turn any of these rules into a worked solution on UnitConv: differentiate a function step by step with the derivative calculator, find antiderivatives and definite integrals with the integral calculator, look up the trigonometric identities you will need along the way, and tidy up large or small numbers with the scientific notation converter.