# Differentiate and find the square of dy/dx

• Dec 5th 2009, 12:45 AM
zorro
Differentiate and find the square of dy/dx
Question:

If $x = cos \theta + \theta sin \theta$ , $y = sin \theta - \theta cos \theta$ , then show that

$\frac{dy}{dx} = tan \theta$

Also find $\frac{d^2 y}{dx^2}$
• Dec 5th 2009, 01:51 AM
mr fantastic
Quote:

Originally Posted by zorro
Question:

If $x = cos \theta + \theta sin \theta$ , $y = sin \theta - \theta cos \theta$ , then show that

$\frac{dy}{dx} = tan \theta$

Also find $\frac{d^2 y}{dx^2}$

$\frac{dy}{dx} = \frac{\frac{dy}{d \theta}}{\frac{dx}{d \theta}}$ (which I'm sure will be somewhere in yur class notes or textbook).

$\frac{d^2y}{dx^2} = \frac{d}{dx} \left(\frac{dy}{dx}\right) = \frac{d}{d \theta} \left(\frac{dy}{dx}\right) \cdot \frac{d \theta}{dx}$

If you need more help, please show all your working and state specifically where you are stuck.
• Dec 5th 2009, 04:14 AM
HallsofIvy
"yur"? (Worried)
• Dec 6th 2009, 07:50 PM
zorro
Is this correct?
Quote:

Originally Posted by mr fantastic
$\frac{dy}{dx} = \frac{\frac{dy}{d \theta}}{\frac{dx}{d \theta}}$ (which I'm sure will be somewhere in yur class notes or textbook).

$\frac{d^2y}{dx^2} = \frac{d}{dx} \left(\frac{dy}{dx}\right) = \frac{d}{d \theta} \left(\frac{dy}{dx}\right) \cdot \frac{d \theta}{dx}$

If you need more help, please show all your working and state specifically where you are stuck.

I am getting the $\frac{d^2 y }{dx^2} = sec^2 \theta .\frac{1}{ \theta cos \theta}$

$
\frac{d}{d \theta} \left( \frac{dy}{dx} \right) = \frac{d}{d \theta} (tan \theta) = sec^2 \theta
$

$
\frac{d \theta}{dx} = \frac{1}{\frac{dx}{d \theta}} = \frac{1}{\theta cos \theta}
$

therefore

$
\frac{d^2 y}{dx^2} = sec^2 \theta . \frac{1}{ \theta cos \theta}

$

Is this Right!!!
• Dec 7th 2009, 12:31 AM
mr fantastic
Quote:

Originally Posted by zorro
I am getting the $\frac{d^2 y }{dx^2} = sec^2 \theta .\frac{1}{ \theta cos \theta}$

$
\frac{d}{d \theta} \left( \frac{dy}{dx} \right) = \frac{d}{d \theta} (tan \theta) = sec^2 \theta
$

$
\frac{d \theta}{dx} = \frac{1}{\frac{dx}{d \theta}} = \frac{1}{\theta cos \theta}
$

therefore

$
\frac{d^2 y}{dx^2} = sec^2 \theta . \frac{1}{ \theta cos \theta}

$

Is this Right!!!

Yes, and if you recall that $\frac{1}{\cos \theta} = \sec \theta$ then you can simplify your answer a little bit.
• Dec 7th 2009, 05:46 PM
zorro
So after simplifying
Quote:

Originally Posted by mr fantastic
Yes, and if you recall that $\frac{1}{\cos \theta} = \sec \theta$ then you can simplify your answer a little bit.

= $sec^2 \theta . \frac{1}{\theta cos \theta}$
= $\frac{sec^3 \theta}{ \theta}$

Is this right now???
• Dec 8th 2009, 12:51 AM
mr fantastic
Quote:

Originally Posted by zorro
= $sec^2 \theta . \frac{1}{\theta cos \theta}$
= $\frac{sec^3 \theta}{ \theta}$

Is this right now???

Correct.
• Dec 8th 2009, 02:19 AM
HallsofIvy
By the way, Zorro, $\frac{d^2y}{dx^2}$ is NOT "the square of dy/dx"!
• Dec 8th 2009, 02:31 AM
zorro
Thanks u every one for helping me
Thank u Mr fantastic for ur help (Bow)