Cheatography

# Calculus Derivatives and Differentiation Cheat Sheet by CROSSANT

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

### Notation

 Name Operation y versions f(x) versions Compos­ition versions Second derivative nth derivative Leibniz 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 Notation d/dx (f(x))­=d/dx (y) y' f'=f'(x)=(f(x))' (f(g(x)))' y'' fn(x) Newton/Dot Notation d/dt (f(t))­=d/dt (y(t)) ẏ ÿ Euler/­D-N­otation Dx(f) Dy Df D(f(g)) D2f Dnf
n ∈ ℕ1 = {1,2,3­­,4­,­5­,...}

### Derivative Rules

 Formal­/Limit Definition of a Derivative lim h->0 (f(x+h­)-f­(x))/h 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≠-m 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/|x| d/dx (|u(x)|), u(x)≠0 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