Derivative of trig function

Well i've managed to handle these pretty well considering I was absolutely stumped during Limits of trig functions. However on a more difficult homework question I came out with an incredibly huge answer which was far from the real one.

Let me try latex for the first time...

[tex]\frac {dy} {dx} \frac {\sin(x)\sec(x)} {1+x\tan(x)} = ?[/tex]

I know the answer is (because of the textbook)

[tex]\frac {1} {(1+x\tan(x))^2}[/tex]

But I came out with a huge answer and i'm stumped. I tried using the product rule for the numerator and then using the quotient rule on the result and the denominator. Keep in mind, we just did those two before doing derivatives of trig functions.

 

Actually the way the question was written was find f'(x) but I like Leibniz better.

Well I erased my work because it confused me more than it aided me :rofl: but i'll give it a try.

I applied the product rule to the numerator (sin(x)sec(x)) using the derivatives of basic trig functions, I got this for the numerator.

[tex] \sin(x)\sec(x)\tan(x) + \sec(x)\cos(x) [/tex]

Looks simple enough but then I applied the quotient rule.

[tex] \frac {\sin(x)\sec(x)\tan(x) + \sec(x)\cos(x) + \sin(x)\sec(x)\tan^2(x) + \sec(x)\cos(x) - \sin(x)\sec^3(x)} {(1+tan(x))^2} [/tex]

Quite far from the answer...

 

Sin(x)/Cos(x)?

Which is Tan(x), therefore i'm an idiot :rofl:

Hold on let me work this out.

 

That would make the equation

[tex] \frac {\tan(x)} {1+\tan(x)} [/tex]

And using the quotient rule I come out with

[tex] \frac {\sec^2(x)+\sec^2(x)\tan(x)-\tan(x)\sec^2(x)} {(1+\tan(x))^2} [/tex]

That would simplify to

[tex] \frac {\sec^2(x)} {(1+\tan(x))^2} [/tex]

Which is not the answer in the back of the book. Should I be getting rid of that 1 + before plugging into the quotient rule? I don't think so because that wouldn't have worked in a similar question which I got correct.

 

Ok I'm going to break something... I swear my brain was here prior to me doing this question.

Would that not give the same result though? Just with an x in front of tanx on the denominator.

 

I differentiated 1+xtan(x) and got xsec^2(x) + tan(x)

Using that I applied the quotient rule and got this

[tex] \frac {\sec^2(x) + x\tan(x)\sec^2(x) - x\sec^2(x)\tan(x) + \tan^2(x)} {(1-x\tan(x))^2} [/tex]

Which simplifies to

[tex] \frac {\sec^2(x) + tan^2(x)} {(1-x\tan(x))^2} [/tex]

Now am I assuming correctly that you can apply the pythagorean identity to this and get an answer of 1/1 on the numerator and the denominator staying the same, giving me my correct answer?

 

Yep I got it all right, and got a nice mark for class participation for explaining it to everyone who didn't get it, which just happened to be the entire class :rofl:

Thanks! Now on to Chain rule homework...