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

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In summary, the student was stuck trying to solve for the derivative of y=sec(ax) using the definition and aproximations. They attempted to do it using the quotient rule, but were not able to get the answer. They then realized that they needed to use the limits, and plugged back into the limit to find the derivative.
  • #1
cpscdave
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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)/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 aproximate 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.
 
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  • #2
cpscdave said:

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).
cpscdave said:
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 + 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.
cpscdave said:
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.

[tex]\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)}[/tex]

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
[tex]\lim_{h \to 0}\frac{sin(ah)}{h} = a[/tex]
and
[tex]\lim_{h \to 0}\frac{1 - cos(ah)}{h} = 0[/tex]
 
  • #3
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
 

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

1. What is the definition of a derivative?

The derivative of a function at a specific point is the slope of the tangent line to the curve of the function at that point.

2. Why is the definition of a derivative useful?

The definition of a derivative allows us to find the instantaneous rate of change of a function at any given point, which is useful in many real-world applications such as physics, economics, and engineering.

3. What is the process for finding the derivative using the definition?

The process involves finding the limit of the slope of the secant line between two points on the function as the distance between those points approaches zero. This will give us the slope of the tangent line and therefore the derivative at that point.

4. How do you apply the definition to find the derivative of y=sec(ax)?

To find the derivative of y=sec(ax), we plug the function into the definition of a derivative and simplify using trigonometric identities. The final result will be the derivative of y=sec(ax) with respect to x.

5. Are there any shortcuts for finding the derivative instead of using the definition?

Yes, there are many derivative rules and formulas that can be used to find the derivative of a function without having to use the definition. These include the power rule, product rule, quotient rule, and chain rule.

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