unit 3: differentiation part 2 - composite, implicit, and inverse functions

derivative of (f◦g)(x) = f(g(x))

f’(g(x)) * (g’(x))

chain rule in words

multiply the derivative of the inside function by the derivative of the outside function

derivative of g(x)ⁿ

(n) * (g(x))^n-1 * (g’(x))

derivative of e^g(x)

g’(x) * e^g(x)

derivative of y

= dy/dx

derivative of x

= 1

inverse functions

f(f^-1(x)) = x = f^-1(f(x))

(a,b) on f(x)

(b,a) on f^-1(x)

existence of an inverse function means

one-to-one and strictly monotonic on entire domain so either increasing whole domain or decreasing whole domain

f’(12) ?

when f(x) = 12

f^-1(x) =

1 / f’(f^-1(x))

derivative of a^x

(ln(a)) * a^x

derivative of a^u

(ln(a)) * (a^u) * (du/dx)

derivative of log_ax

1 / (ln(a) * x)

derivative of log_au

( 1 / (ln(a) * u) ) * (du/dx) = u’ / (ln(a)*x)

common log =

log₁₀

a*ln(b) is the same as

ln(b)^a

derivative of arcsin u

( 1 / sqrt(1-u²) ) * u’

derivative of arccos u

( -1 / sqrt(1-u²) ) * u’

derivative of arctan u

( 1 / 1+u² ) * u’

derivative of arccsc u

( -1 / |u|*sqrt(u²-1) ) * u’

derivative of arcsec u

( 1 / |u|*sqrt(u²-1) ) * u’

derivative of arccot u

( -1 / (1+u²) ) * u’