# Unit 3 - Differentiation Implicitly, and with Compositions and Inverses

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What is Chain Rule and what is it used for?

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AB Calculus students!!

### 17 Terms

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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

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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

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Can you take the log of a negative number?

NO!!

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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

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5

d/dx(eu) = ?

eu(u’)

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What is the base change formula for exponents and logs?

• ax = (eln(a))x

• logax = (1/(lna))(lnx)

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Derivative of |x|?

x/|x|

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derivative of arcsin(x) or arccos(x)

±1/sqrt(1-x²)

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derivative of arcsec(x) or arccsc(x)

±1/sqrt((x²)(x²-1))

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derivative of arctan(x) or arccot(x)

1/1+x²

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the derivative of f-1(y of f)

1/f’(x)

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s(t)

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

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s’(t)

velocity (distance/time)

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s’’(t)

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

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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)

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derivative of ax

ax*ln(a)

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derivative of log base a (x)

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

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