Re: Overlapping Chain Rule?

Quote:

Originally Posted by

**Biff** find the derivative:

$\displaystyle y = \sin^2(\cos(3x^{-2}))$

Try

$\displaystyle \left\{ {2\sin \left( {\cos \left( {3x^{ - 2} } \right)} \right)} \right\}\left[ {\cos \left( {\cos \left( {3x^{ - 2} } \right)} \right)} \right]\left[ { - \sin \left( {3x^{ - 2} } \right)} \right]\left[ { - 6x^{ - 3} } \right]$

Re: Overlapping Chain Rule?

You can apply the chain rule as many times as you have strength for! Here, you have $\displaystyle y(x)= sin^2(cos(3x^{-2}))$

Let $\displaystyle u= 3x^{-2}$. Then $\displaystyle y(u)= sin^2(cos(u))$. Let v= cos(u). Then $\displaystyle y(v)= sin^2(v)$. Let $\displaystyle w= sin(v)$. Then $\displaystyle y(w)= w^2$.

Then $\displaystyle \frac{dy}{dx}= \frac{dy}{dw}\frac{dw}{dv}\frac{dv}{du}\frac{du}{d x}$ which gives what Plato said.

Re: Overlapping Chain Rule?

Now this makes sense - thanks guys!

Here's how I solved it (separating it out helped):

$\displaystyle \frac{dy}{dx} = 2\sin(\cos(3x^{-2})) \cdot \cos(\cos(3x^{-2})) \cdot (-\sin(3x^{-2})) \cdot -6x^{-3}$

$\displaystyle \frac{dy}{dx} = 12x^{-3}\sin(\cos(3x^{-2})) \cdot \cos(\cos(3x^{-2})) \cdot \sin(3x^{-2}))$