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Calculus Derivatives and Differentiation Cheat Sheet

A Education › Mathematics cheat sheet in English by CROSSANT

Derivatives rules and common derivatives from Single-Variable Calculus. Most important rules/derivatives are bolded.

Function Calculus Df Rule Chain Dx Constant Derivative Dy Gamma Nth
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Notation

Name
Operation
y versions
f(x) versions
Compos­ition versions
Second derivative
nth derivative
Leibni­z/F­raction Notation
d/dx (f(x))­=d/dx (y)
dy/dx=dy(x)/dx
df/dx=­df(­x)/­dx=­d(f­(x))/dx
df/dg*­dg/dx
d2f/dx2
dn/dxn=dnf/dxn
Lagran­ge/­Prime Notation
d/dx (f(x))­=d/dx (y)
y'
f'=f'(x)=(f(x)­)'
(f(g(x­)))­'
f''
f(n)(x)
Newton/Dot Notation
d/dt (f(t))­=d/dt (y(t))
ẏ
    
ÿ
Euler/­D-N­otation
Dx(f)
Dxy
Dxf
Dx(f(g))
Dx2f
Dxnf
n ∈ ℕ1 = {1,2,3­­,4­,­5­,...}

Derivative Rules

Formal­/Limit Definition of a Derivative
f'(x­)=lim h->0 (f(x+h­)-f­(x))/h
Limit Definition of the Derivative at a point
f'(a­)=lim h->0 (f(a+h­)-f­(a))/h
f'(a­)=lim x->a (f(x)-­f(a­))/­(x-a)
Linearity 1: Consta­nt-­Mul­tiple Rule
d/dx (kf(x))
k*d/dx (f)
kf'
Linearity 2: Sum/Di­ffe­rence Rule
d/dx (f(x)±­g(x))
d/dx (f) ± d/dx (g)
f'±g­'
Product Rule
d/dx (f(x)*­g(x))
f'g+­fg'
Multi-­Product Rule
d/dx (p(x)*­q(x­)*r­(x)­*s(­x)*...)
p'qrs... + pq'rs... + pqr's... + pqrs'... + ...
pqrs...*(­p'/p + q'/q + r'/r + s'/s + ...)
Quotient Rule
d/dx (f(x)/­g(x))
(f'g­-fg­')/g2
g(x)≠0, quotients can be rewritten into products with sign-f­lipped exponents
Chain Rule
d/dx (f(g(x)))
f'(g­)g'
Multi-­Chain Rule
d/dx (p(q(r­(s(...)­))))
p'(q­(r(­s(...)­)))­*q'(­r(s­(...))­)*r­'­(s(...)­)*­s'(...)*...
First Fundam­ental Theorem of Calculus (FTC I)
d/dx (∫ax f(t)dt)
f(x)
Deriva­tives and integrals are inverses of each other
FTC I Chain Rule 1
d/dx (∫av(x) f(t)dt)
f(v)v'
FTC I Chain Rule 2
d/dx (∫u(x)v(x) f(t)dt)
f(v)v'-­f(u­)u'
Summation Rule
d/dx (Σf(x))
Σf'(x)
The summation must be within its interval of conver­gence
a and k are constants
f, g, p, q, r, s, u, and v are functions of x such that f=f(x), g=g(x), p=p(x), q=q(x), r=r(x), s=s(x), u=u(x), and v=v(x), unless otherwise shown

Deriva­tives of Algebraic Functions

Rule
Function Derivative
Derivative
nth Derivative of Function
nth Derivative
Function Compos­ition
Derivative by Chain Rule
Constant
d/dx (k)
0
dn/dxn (k)
0
d/dx (f(k))
0
Power
d/dx (xk)
kxk-1
dn/dxn (xk), k≠0, k-n+1≠-n
Γ(k+1)xk-n/Γ(k-n+1)
d/dx (u(x)k), u(x)≠0
kuk-1u'
Natural Expone­ntial
d/dx (ex)
ex
dn/dxn (ex)
ex
d/dx (eu(x))
euu'
Natural Logarithm
d/dx (ln(x))
1/x
dn/dxn (ln(x))
(-1)n+1(n-1)!/xn
d/dx (ln(u(x))), u(x)>0
u'/u
General Expone­ntial
d/dx (kx), k>0
kxln(k)
dn/dxn (kx), k>0
kx(ln(k))n
d/dx (ku(x)), k>0
kuln(k)u­'
General Logarithm
d/dx (logk(x)), k>0, k≠1
1/(xln(k))
dn/dxn (logk(x)), k>0, k≠1
(-1)n+1(n-1)!/(xnln(k))
d/dx (logk(u(x))), k>0, k≠1, u(x)≠0
u'/(­uln(k))
Absolute Value
d/dx (|x|), x≠0
x/|x| OR |x|/x
    
d/dx (|u(x)|), u(x)≠0
u*u'/|u| OR |u|*u'/u
Functi­on-­Pow­er-­Fun­ction
d/dx (f(x)g(x)), f(x)>0
fg(f'g­/f+­ln(­f)g­')
k is a constant
f=f(x), g=g(x), and u=u(x) are all functions of the variable x
m, n ∈ ℕ1 = {1,2,3­,4,­5,...}
Γ(x) is the gamma function, which defines factorials for negati­ve/­non­-in­teger numbers
x! = Γ(x+1)
n! = n(n­-1­)­!­=n­­(n-­­1)­(­n­-2­­)!=­­n(­n­-­1)­­(n-­­2)­(­n­-3­­)!=...
n! = n(n-1)­­(n­-­2­)(­­n-3­­)...*­3*2*1
0!=1, 1!=1

Deriva­tives of Trigon­ometric Functions

Standard Trigon­ometric
Derivative
Inverse Trigon­ometric
Derivative
Hyperbolic Trigon­ometric
Derivative
Hyperbolic Inverse Trigon­ometric
Derivative
d/dx (sin(x))
cos(x)
d/dx (arcsi­n(x))
1/√(1-x2)
d/dx (sinh(x))
cosh(x)
d/dx (arcsi­nh(x))
1/√(1+x2)
d/dx (cos(x))
-sin(x)
d/dx (arcco­s(x))
-1/√(1-x2)
d/dx (cosh(x))
sinh(x)
d/dx (arcco­sh(x))
-1/√(x2-1)
d/dx (tan(x))
sec2(x)
d/dx (arcta­n(x))
1/(1+x2)
d/dx (tanh(x))
sech2(x)
d/dx (arcta­nh(x))
1/(1-x2)
d/dx (csc(x))
-csc(x­)cot(x)
d/dx (arccs­c(x))
-1/(|x|√(x2-1))
d/dx (csch(x))
-csch(­x)c­oth(x)
d/dx (arccs­ch(x))
-1/(|x|√(x2+1))
d/dx (sec(x))
sec(x)­tan(x)
d/dx (arcse­c(x))
1/(|x|√(x2-1))
d/dx (sech(x))
-sech(­x)t­anh(x)
d/dx (arcse­ch(x))
-1/(|x­|√(1-x2))
d/dx (cot(x))
-csc2(x)
d/dx (arcco­t(x))
-1/(1+x2)
d/dx (coth(x))
-csch2(x)
d/dx (arcco­th(x))
1/(1-x2)
dn/dxn (sin(x)) = sin(x+­nπ/2)
dn/dxn (cos(x)) = cos(x+­nπ/2)
sinh(x) = (ex-e-x)/2
cosh(x) = (ex+e-x)/2
arcsinh(x) = ln(x+√(x2+1))
arccosh(x) = ln(x+√(x2-1)), x≥1

Polynomial Derivative Examples

d/dx (x)
1
d/dx (x^2)
2x
d/dx (x^3)
3x2
d/dx (x^4)
4x3
d/dx (1/x)
-1/x2
d/dx (-1/x2)
2/x3
d/dx (2/x3)
-6/x4
d/dx (-6/x4)
24/x5
d/dx (√x)
1/(2√x)
d/dx (x1/3)
1/(3x2/3)
d/dx (x1/4)
1/(4x3/4)
d/dx (x3/2)
3√x/2
d/dx (x5/3)
5x2/3/3
d/dx (x-√2-3)
(-√2-3)x-√2-4
d/dx (1/(1+x))
-1/(1+x)2
d/dx (-1/(1+x)2)
2/(1+x)3
d/dx (-1/(1-x))
-1/(1-x)2
d/dx (-1/(1-x)2)
-2/(1-x)3
d/dx (√(5x+1))
5/(2√(­4x+1))
d/dx (√(x5+1))
5x4/(2√(x5+1))
d/dx ((2x2+5)9)
36x(2x2+5)8
d/dx (1)
0

Specia­l/Other Derivative Examples

d/dx (exsin(x))
exsin(x)+excos(x)
d/dx (excos(x))
excos(x)-exsin(x)
d/dx (sinx(x))
sinx(x)(ln­(si­n(x­))+­xco­t(x))
d/dx (sin(x)cos(x))
sin(x)cos(x)(cos2(x)csc­­(x­)­-­si­­n(x­­)l­n­(­si­­n(x)))
d/dx (ln(1/­(1-x)))
1/(1-x)
d/dx (ln(x3+7x+12))
(3x2+7)/(x3+7x+12)
d/dx (ln(e3xtan(x3)))
3+(3x2sec2(x3))/(tan(x3))
d/dx (1+k+t­+√2­+co­s(a­)+e­+π+­ln(3))
0

Trigon­ometric Derivative Examples

d/dx (-sin(x))
-cos(x)
d/dx (-cos(x))
sin(x)
d/dx (sin(2x))
2cos(2x)
d/dx (cos(2x))
-2sin(2x)
d/dx (sin2(x))
2sin(x­)cos(x)
d/dx (cos2(x))
-2cos(­x)s­in(x)
d/dx (arcta­n(3x))
3/(1+9x2)
d/dx (sin(s­in(x)))
cos(x)­cos­(si­n(x))
d/dx (sin(a­rcc­os(x)))
-x/√(1-x2)
d/dx (sin(k))
0

Expone­ntial Derivative Examples

d/dx (xex)
ex+xex
d/dx (e2x)
2e2x
d/dx (ex²)
2xex²
d/dx (eeˣ)
exeeˣ
d/dx (xx)
xx(ln(x)+1)
d/dx (23ˣ)
23ˣ*3x*ln(2)­*ln(3)
d/dx (ek)
0

Logari­thmic Derivative Examples

d/dx (ln(1/x))
-1/x
d/dx (ln(1+x))
1/(1+x)
d/dx (ln(1-x))
-1/(1-x)
d/dx (ln(x2))
2/x
d/dx (ln(x3))
3/x
d/dx (ln(x4))
4/x
d/dx (xln(x))
ln(x)+1
d/dx (ln(ln­(x)))
1/(xln(x))
d/dx (ln(k))
0

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