smc-736-60-Chain Rule

Calculus, MET1 2019 VCAA 1b

Let `g: R text(\ {−1}) -> R,\ \ g(x) = (sin(pi x))/(x + 1)`.

Evaluate `g^{prime}(1)`. (2 marks)

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`-pi/2`

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	`u = sin(pi x)`	`v = x + 1`
	`u^{prime}=pi cos(pi x)`	`v^{prime}=1`

Let `f(x)=(1 + tan x)^10.` Find `f^{\prime}(x)`. (2 marks)

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`10 sec^2 x \ (1 + tan x)^9`

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`f(x) = (1 + tan x)^10`

`f^{\prime}(x)`	`= 10 (1 + tan x)^9 xx d/(dx) (tan x)`
	`= 10 sec^2 x \ (1 + tan x)^9`

If `f(x)= 2 sin 3x - 3 tan x`, find `f^{prime}(0)`. (2 marks)

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`3`

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`y`	`= 2 sin 3x-3 tan x`
`(dy)/(dx)`	`= 6 cos 3x-3 sec^2 x`

`text(When)\ \ x = 0,`

Let `g(x) = log_e(tan(x))`. Evaluate `g^{prime}(pi/4)`. (2 marks)

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`g^{prime}(pi/4) = 2`

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`g(x) = log_e (tan(x))`

`text(Using Chain Rule:)`

`g^{prime}(x) = (sec^2(x))/(tan(x))`

`text(When)\ \ x = pi/4,`