Unit 3 - Differentiation Implicitly, and with Compositions and Inverses

• h(x) = f(g(x)), then h’(x) = f’(g(x))(g’(x))

• OR dy/dx = dy/du(du/dx)

• Finding the derivative of compositions

A shortcut can be used! d/dx(lnu), where u is a function of x, = u’/u

NO!!

• ln(0) = 1

• ln(ab) = lna + lnb

• (ln(a)^n) = nln(a) → notice how n is on the inside of the parentheses!

• ln(a/b) = lna - lnb

e^u(u’)

• a^x = (e^ln(a))^x

• log_ax = (1/(lna))(lnx)

x/|x|

±1/sqrt(1-x²)

±1/sqrt((x²)(x²-1))

1/1+x²

1/f’(x)

position function - postive velocity means the object moves up/forward/right; negative velocity means object moves down/backward/left

velocity (distance/time)

acceleration (distance/time²) or (distance/time)/time

1. Differentiate in regards to x

2. Seperate the y’ terms from the non-y’ terms

3. Isolate y’

(sometimes putting in values first is easier)

a^x*ln(a)

1/xln(a) or 1/x * 1/lna