Unit 3 - Differentiation Implicitly, and with Compositions and Inverses

AB Calculus students!!

What is Chain Rule and what is it used for?

• 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

What if I have a composition with ln?

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

Can you take the log of a negative number?

NO!!

What are the log properties?

• 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

d/dx(e^u) = ?

e^u(u’)

What is the base change formula for exponents and logs?

• a^x = (e^ln(a))^x

• log_ax = (1/(lna))(lnx)

Derivative of |x|?

x/|x|

derivative of arcsin(x) or arccos(x)

±1/sqrt(1-x²)

derivative of arcsec(x) or arccsc(x)

±1/sqrt((x²)(x²-1))

derivative of arctan(x) or arccot(x)

1/1+x²

the derivative of f^-1(y of f)

1/f’(x)

s(t)

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

s’(t)

velocity (distance/time)

s’’(t)

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

Implicit Differentiation

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)

derivative of a^x

a^x*ln(a)

derivative of log base a (x)

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