Find the Derivative of y=sec(ax) using defition

[cpscdave]

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)/cos²(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 approximate sin(ah) and cos(ah)
Linear aprox of sin(ah) = ah + ah³/6
Linear aprox of cos(ah) = 1 + ah²/2

this gives us
cos(ax)(1 + ah²/2) - sin(ax)(ah + ah³/6

cos(ax) + cos(ax)ah²/2 -ah*sin(ax) - sin(ax)ah³/6

plug back into limit
(cos(ax) - cos(ax) kills that)
lim 1 / cos(ax)ah²/2 -ah*sin(ax) - sin(ax)ah³/6 / h
h->0

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

 

[Mark44]

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)/cos²(ax)

Actually, it's a times the above, or a sec(ax)tan(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 approximate sin(ah) and cos(ah)
Linear aprox of sin(ah) = ah + ah³/6
Linear aprox of cos(ah) = 1 + ah²/2

this gives us
cos(ax)(1 + ah²/2) - sin(ax)(ah + ah³/6

cos(ax) + cos(ax)ah²/2 -ah*sin(ax) - sin(ax)ah³/6

plug back into limit
(cos(ax) - cos(ax) kills that)
lim 1 / cos(ax)ah²/2 -ah*sin(ax) - sin(ax)ah³/6 / h
h->0

This is very hard to read. You have four / signs indicating division, so it's difficult to determine what's in the numerator and what's in the denominator.

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

\frac{1}{h}\cdot \frac{cos(ax) - cos(ax)cos(ah) + sin(ax)sin(ah)}{(cos(ax)cos(ah) - sin(ax)sin(ah))cos(ax)}

To see my LaTeX script, click on the expression above.

Next, write this as two fractions, using the first two terms in the numerator for the first fraction, and the third term for the other fraction.

Another approach besides approximating sin(ah) and cos(ah) is to work with the limits
\lim_{h \to 0}\frac{sin(ah)}{h} = a
and
\lim_{h \to 0}\frac{1 - cos(ah)}{h} = 0

 

[cpscdave]

Perfect that clued me in exactly where I was going wrong.

Never occurred to me in the first step to put sec(ax+ah) and sec(ax) over a common denominator.

Thanks :D