Derivatives Test

Definition of a Derivative
f ' (x) =    lim     f (x + Δx) – f (x)

Δx->0            Δx

Alternative Form
f ' (c) =    lim     f (x) – f (c)

Δx->c       x – c

Rates of Change
Average Velocity
Δs = f (x2) – f (x1)
Δt        x2 – x1

Instantaneous Velocity
ds = f '(x1), f '(x2) (one answer per time given)
dt

Power Rule
d/dx [xn] = nxn – 1

Trig Derivatives
d/dx [sin x] = cos x d/dx [cos x] = –sin x
d/dx [tan x] = sec2 x d/dx [cot x] = –csc2 x
d/dx [sec x] = sec x tan x d/dx [csc x] = –csc x cot x

Product rule
Formula:

f(x) = f (g)f (h)
f '(x) = f '(g)f (h) + f (g)f '(h)

Quotient rule
Formula:

f (x) = f (g)
f (h)
f '(x) = f (h)f '(g) – f (g)f '(h)
(f (h))2

Chain Rule
Composite  Regular
√(1 + x2)   1 + x2
sin 2x   sin x

Formula:

y = f ( g(x) )
y ' = f '( g(x) ) g '(x)

Eg.
y = sinx2
y ' = (cos x2) (2x)

Implicit Differentiation
Differentiate (y) as you differentiate (x), except an additional (y ') is multiplied.
Eg.

y = 2y2
y '= (2)(2y)y '

Related Rates
Write formula, differentiate formula, plug in givens, solve.