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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,...}
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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
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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
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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
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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/(2√x) |
d/dx (x5/3) |
5/(3x2/3) |
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 |
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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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