derivatives Flashcards
Constant
Function: C
Derivative: 0
Line
Function: x Derivative: 1
Function: ax Derivative: a
Square
Function: x^2
Derivative: 2x
Square Root
Function:√x
Derivative: (½)x-½
Exponential
Function: e^x Derivative: e^x
Function: a^x Derivative: ln (a) a^x
Logarithms
Function: ln(x) Derivative: 1/x
Function: loga(x) Derivative: 1 / (x ln(a))
TRIG sin(x)
Function: sin(x)
Derivative: cos(x)
TRIG cos(x)
Function: cos(x)
Derivative: -sin(x)
TRIG tan(x)
Function: tan(x)
Derivative: sec^2 (x)
Rules: Multiplication by constant
F: cf D: cf’
Power Rule
F:x^n
D: nx^n−1
Sum Rule
F: f + g
D: f’ + g’
Difference Rule
F: f - g
D: f’ − g’
Product Rule
F: fg
D: f g’ + f’g
Quotient Rule
F: f/g
D: (f’ g − g’ f )/g^2
Reciprocal Rule
F: 1/f
D: −f’/f^2
Chain Rule (as composition of functions)
F: f º g
D: (f’ º g) × g’
Chain Rule (using ‘)
F: f(g(x))
D: f’(g(x))g’(x)
Chain Rule ((using d/dx )
F: dy/dx = dy/du * du/dx
which just means derivative of the whole x derivative of the inside
d/dx
“The derivative of” is also written d/ dx
Exponentiation: Fractional Exponents
x^m/n = the n root of x^m or (n root of x)^m
Exponentiation: Power of a product
(xy)^a=x^ay^a
natural log Product Rule
ln(xy)=ln(x)+ln(y)
natural log Quotient Rule
ln(x/y)=ln(x)−ln(y)
Log of e
ln(e)=1
Log of one
ln(1)=0
Log reciprocal
ln(1/x)=−ln(x)
