average rate of change

average rate of change

Δy f (x2 ) − f (x1 ) = Δx x2 − x1

definition of a derivative:

f '(x) =

definition of a derivative: instantaneous rate of change f (x + h) − f (x) f '(x) = lim h→0 h

Constant Rule

d c= dx

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Constant Rule

d c=0 dx

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Constant Multiple Rule

d cf (x) = dx

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Constant Multiple Rule

d cf (x) = cf ′(x) dx

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Power Rule

d n x = dx

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Power Rule

d n n −1 x = nx dx

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Product Rule

d f (x)ig(x) = dx

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Product Rule

d f (x)ig(x) = f (x)i g′(x) + g(x)i f ′(x) dx

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Quotient Rule

d f (x) = dx g(x)

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Quotient Rule d f (x) [ g(x)i f ′(x)] − [ f (x)i g′(x)] = 2 dx g(x) [ g(x)]

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d/dx sinx

d sin x = dx

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d/dx sinx

d sin x = cos x dx

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d/dx cosx

d cos x = dx

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d/dx cosx

d cos x = − sin x dx

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d/dx tanx

d tan x = dx

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d/dx tanx

d 2 tan x = sec x dx

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d/dx cscx

d csc x = dx

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d/dx cscx

d csc x = − csc x cot x dx

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d/dx secx

d sec x = dx

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d/dx secx

d sec x = sec x tan x dx

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d/dx cotx

d cot x = dx

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d/dx cotx

d 2 cot x = − csc x dx

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Chain Rule (dx notation)

dy dx

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Chain Rule (dx notation)

dy dy du = i dx du dx

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Chain Rule (f ’ notation)

d f ( g(x)) = dx

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Chain Rule (f ’ notation)

d f ( g(x)) = f ′ ( g(x))i g′(x) dx

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Exponential Functions

d x e = dx

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Exponential Functions

d x x e =e dx

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Exponential Functions

d u e = dx d g( x ) e = dx d g( x ) a = dx

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Exponential Functions

d u u du e =e dx dx d g( x ) g( x ) e = e i g′(x) dx d g( x ) g( x ) a = a i g′(x)iln a dx 34

Logarithmic Functions

d ln x = dx

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Logarithmic Functions

d 1 ln x = , x > 0 dx x

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Logarithmic Functions

d ln u = dx d ln g(x) = dx d log b g(x) = dx

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Logarithmic Functions

d 1 du ln u = i dx u dx d 1 g′(x) ln g(x) = i g′(x) = dx g(x) g(x) d g′(x) log b g(x) = dx ln bg(x) 38

Derivative of Inverse Functions

( )

If f (a) = b, then f

−1

′ (b ) =

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Derivative of Inverse Functions

( )

If f (a) = b, then f

−1

′ (b ) =

1 f ′(a)

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d/dx sin-1x

d −1 sin x = dx

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d/dx sin-1x

d 1 −1 sin x = 2 dx 1− x

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d/dx cos-1x

d −1 cos x = dx

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d/dx cos-1x

d −1 −1 cos x = 2 dx 1− x

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d/dx tan-1x

d −1 tan x = dx

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d/dx tan-1x

d 1 −1 tan x = 2 dx 1+ x

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d/dx cot-1x

d −1 cot x = dx

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d/dx cot-1x

d −1 −1 cot x = 2 dx 1+ x

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d/dx sec-1x

d −1 sec x = dx

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d/dx sec-1x

d 1 −1 sec x = 2 dx | x | x −1

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d/dx csc-1x

d −1 csc x = dx

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d/dx csc-1x

d −1 −1 csc x = 2 dx | x | x −1

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