Chapter 2 Flashcards
Difference quotient
(f (x+h) - f (x)) / h
Derivative
The instantaneous rate of change of a function w/ respect to another variable
The slope of the tangent to f at x = c
f ‘ (c) = the derivative of the function at that particular point
Common derivative notation
f ‘ (x) or d / dx
d/dx (sin x)
cos x
d/dx (cos x)
-sin x
d/dx (e^x)
e^x
d/dx (ln x)
1/x
Derivative of a constant (d/dx (c))
= 0
d/dx (1/x)
-1/x^2 or -x^-2 ….but -1/x^2 is preferred
d/dx ( sq rt (x))
1 / 2 sq rt x
Equation for a line through (x_1, y_1) with slope ‘m’
y - y_1 = m (x - x_1)
a position function
s(t)
a velocity
v(t)…..which equals s’(t) = derivative of position function
an acceleration
a(t)…..which equals v’(t) and s’‘(t) = derivative of a velocity function and second derivative of position function
d/dx (tan x)
sec² x
d/dx (sec x)
sec x tan x
d/dx (cot x)
-csc² x
d/dx (csc x)
-csc x cot x
The chain rule
(Derivative of outer function) x (derivative of inner function)
y’ = f’ (g(x)) times g’ (x)
Generalized power rule
d/dx (g(x)^n) = n (g(x))^n-1 times g’ (x)
Quotient Rule
(f/g)’ = (f ‘g - f g’) / g²
or
d/dx (f(x)/g(x)) = (g(x) f ‘(x) - f(x) g’(x)) / [g(x)]²
Product Rule
(fg)’ = f ‘g + fg’
(fgh)’ = f ‘gh + fg’h + fgh’
etc.
