Derivative of a Parametric Function

**nic42991** · October 5th 2008, 07:26 AM

Let $x=t^2+t$ , and $y=sin(t)$

Find $d/dx(dy/dx)$

Thanks

**skeeter** · October 5th 2008, 07:46 AM

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

$\frac{d}{dx}\left[\frac{dy}{dx}\right] = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\fra c{dx}{dt}}$

**nic42991** · October 5th 2008, 08:25 AM

I don't understand how that works out. I understand $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ , but I don't understand this part:

I don't have the intuition behind it.

**Moo** · October 5th 2008, 08:39 AM

Originally Posted by nic42991

I don't understand how that works out. I understand $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ , but I don't understand this part:

I don't have the intuition behind it.

If you prefer, you can say that :

$\frac{df}{dx}=\frac{\frac{df}{dt}}{\frac{dx}{dt}}$ (1)

Now, let $f=\frac{dy}{dx}$ in (1), this gives :

$\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\fra c{dx}{dt}}<br />$

And then transform $\frac{dy}{dx}$ , by letting f=y in (1)

**skeeter** · October 5th 2008, 08:43 AM

let's look at it another way ...

let $\frac{dy}{dt} = y'(t)$ and $\frac{dx}{dt} = x'(t)$

$\frac{d}{dx}\left[\frac{y'(t)}{x'(t)}\right] =$

using the quotient and chain rule ...

$\frac{x'(t) \cdot y''(t) \cdot \frac{dt}{dx} - y'(t) \cdot x''(t) \cdot \frac{dt}{dx}}{[x'(t)]^2} =$

$\frac{x'(t) \cdot y''(t) - y'(t) \cdot x''(t)}{[x'(t)]^2 \cdot \frac{dx}{dt}} =$

$\frac{x'(t) \cdot y''(t) - y'(t) \cdot x''(t)}{[x'(t)]^3} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\fra c{dx}{dt}}$

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