- #1

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## Homework Statement

I need to show the derivative of y=sec(ax) using the definition and aproximations

(Hopefully I'll type this so its understandable what I've done)

## Homework Equations

I realize the answer is sin(ax)/cos

^{2}(ax)

I also can do get the answer using the quotient rule but we're supposed to do it using aprox and definitons

## The Attempt at a Solution

Lim

__Sec(a(x+h) - sex(ax)__

h->0 h

sec(ax+ah) = 1/cos(ax+ah)

cos(ax+ah) = cos(ax)cos(ah)-sin(ax)sin(ah)

ah as h->0 is small so we can aproximate sin(ah) and cos(ah)

Linear aprox of sin(ah) = ah + ah

^{3}/6

Linear aprox of cos(ah) = 1 + ah

^{2}/2

this gives us

cos(ax)(1 + ah

^{2}/2) - sin(ax)(ah + ah

^{3}/6

cos(ax) + cos(ax)ah

^{2}/2 -ah*sin(ax) - sin(ax)ah

^{3}/6

plug back into limit

(cos(ax) - cos(ax) kills that)

lim 1 / cos(ax)ah

^{2}/2 -ah*sin(ax) - sin(ax)ah

^{3}/6 / h

h->0

this is where I'm stuck. I know there some identity or math trickery thats needed but I have no idea where to proceed.