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Level 606

### Derivative Functions

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lim x→a f(x) = f(a)
Continuous function (that f is continuous at a)
= 0
d/dx (c) = ?
= 1
d/dx (x) = ?
= nx^(n-1)
d/dx (x^n) = ?
= cf'(x)
d/dx (cf(x)) = ?
= f'(x)+g'(x)
d/dx (f(x)+g(x)) = ?
= f'(x)-g'(x)
d/dx (f(x)-g(x)) = ?
= e^x
d/dx (e^x) = ?
= f'(x)g(x)+f(x)g'(x)
d/dx [f(x)g(x)] = ?
d/dx [f(x)/g(x)] = ?
= [f'(x)g(x)-f(x)g'(x)]/g(x)^2 or [(d of nom)(denom)-(nom)(d of denom)]/(denom)^2
= sinθ/θ
lim θ→0 of ? = 1
= (cosθ-1)/θ
lim θ→0 of ? = 0
= cos(x)
d/dx sin(x) = ?
= -sin(x)
d/dx cos(x) = ?
= sec(x)^2
d/dx tan(x) = ?
= (secx)(tanx)
d/dx sec(x) = ?
= -csc(x)^2
d/dx cot(x) = ?
(-cscx)(cotx)
d/dx csc(x) = ?
(g°f)'(a) =
= g'(f(a))(f'(a))
d/dx [g(x)]^n = ?
= n[g(x)]^(n-1) * g'(x)
= e^(xlna)
a^x = ?
d/dx a^x = ?
= a^x * lna
= (1/√(1-x^2))
d/dx (sinx)^-1 = ?
= -(1/√(1-x^2))
d/dx (cosx)^-1 = ?
= (1/(1+x^2))
d/dx (tanx)^-1 = ?
= -(1/(x√((x^2)-1)))
d/dx (cscx)^-1 = ?
= (1/(x√((x^2)-1)))
d/dx (secx)^-1 =?
= -(1/(1+x^2))
d/dx (cotx)^-1 = ?
= 1/x
d/dx lnx = ?
= |x|/x
d/dx |x| = ?
= g'(x)/g(x)
d/dx [lng(x)] = ?
e =
lim x→0 (1+x) ^ (1/x)
sec^2 x
f '(x) of tanx
-csc^2 x
d/dx (cot x)
secxtanx
f '(x) of secx
-cscxtanx
f '(x) of cscx
sin' x
cos x
tan' x
sec² x
sec' x
sec x tan x
cos' x
-sin x
cot' x
-csc² x
csc' x
-csc x cot x
H(x)=f(g(x))
H'(x)=g'(x)+f'(g(x))
f(x)=lnx
(Exponential Decay flipped over x-axis)
F(x)=b^x
The exponential function with base b
f(x)=e^x
(Steeper Exponential Growth)
F(x)=sinx
F'(x)=cosx
F(x)=cosx
F'(x)=-sinx
F(x)=tanx
F'(x)=sec^2x
F(x)=cscx
F'(x)=(-cotx)(cscx)
F(x)=secx
F'(x)=(tanx)(secx)
F(x)=cotx
F'(x)=-csc^2x