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Calculus

Derivatives & Differentiation

AP Calculus AB

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

AB Calculus students!!

1

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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2

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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3

Can you take the log of a negative number?

NO!!

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4

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(e^{u}) = ?

e^{u}(u’)

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6

What is the base change formula for exponents and logs?

a

^{x}= (e^{ln(a)})^{x}log

_{a}x = (1/(lna))(lnx)

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7

Derivative of |x|?

x/|x|

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8

derivative of arcsin(x) or arccos(x)

±1/sqrt(1-x²)

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9

derivative of arcsec(x) or arccsc(x)

±1/sqrt((x²)(x²-1))

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10

derivative of arctan(x) or arccot(x)

1/1+x²

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11

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

1/f’(x)

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12

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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13

s’(t)

velocity (distance/time)

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14

s’’(t)

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

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15

Implicit Differentiation

Differentiate in regards to x

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

Isolate y’

(sometimes putting in values first is easier)

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16

derivative of a^{x}

a^{x}*ln(a)

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17

derivative of log base a (x)

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

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