Derivatives Flashcards
d/dx [sinx]
What is the derivative of sinx?
cos x
d/dx [cosx]
What is the derivative of cosx?
- sinx
d/dx [tan x]
What is the derivative of tan x?
1/cos2x = sec2x
d/dx [ex]
ex
d/dx [ln x]
What is the derivative of ln x?
1/x = x-1
d/dx [f(x) g(x)] =
Product rule
d/dx [f(x) g(x)] = f’(x) * g(x) + f(x) *g’(x)
d/dx [f(x)/g(x)] =
Quotient Rule
d/dx [f(x)/g(x)] = [f’(x) g(x) - f(x) g’(x)] / [g(x)]2
What is the relationship between differentiability and continuity?
Differentiability implies continuity
If f is differentiable at x=c, then f is continuous at x=c
Continuity doesn’t however imply differentiability.
What is the expression for the limit method of finding the derivative of a function (the formal form)?
f’(x) = lim h->0 [f(x+h) - f(x)] / h
What is the expression for the alternate form of finding the derivative?
f’(c) = lim x->c [f(x) -f(c)] / x-c
When should you use the formal and alternate form of the derivative?
You should use the formal when you need a general expression for any given point on the line. Use the alternative when you simply have a point at which you need the derivative.
What is the constant rule?
f(x) = C
f’(x) = 0
The derivative of any constant is 0 (zero slope - horizontal line)
(ex: f(x) = 7 , f’(x) = 0)
What is the Power Rule?
f(x) = xn
f’(x) = n*xn-1
Example: f(x) = x5
f’(x) = 5x4
What is the Sum and Difference Rule?
d/dx [f(x) +or- g(x)] = f’(x) +or- g’(x)
Example: p(x) = 3x4 + 2x
p’(x) = 12x3 + 2
RATE OF CHANGE
What is the relationship between the position function - s(t) - and the velocity function v(t)?
The velocity function is the derivative of the position function. It can be viewed at the limit of s(t) as t->0. So the velocity is the slope of the line on a graph (where the x axis is time and y axis is position).
