Using the Chain Rule to solve? Also, find the derivative?

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March 23rd 2009, 06:04 PM
anita123

Using the Chain Rule to solve? Also, find the derivative?

I am confused on what to do with the chain rule. I have a problem that goes a bit like this:

Assume that x, y, and z are all functions of time, t. Use the chain rule to find dz/dt, in terms of x, y, dx/dt and dy/dt if z= sin(x y).

How do I use the chain rule to solve this? Please help me solve and tell me the steps you used to get to it. I have this so far: dz/dt = cosx dy/dt y dx/dt, but I'm pretty sure that's wrong.

Also, I'm confused on how to find the derivative for this one. I get kind of confused when there are squares along with it:

d/dx [sin(x^2)+cos^2(x)].

Any help GREATLY appreciated! Thanks so much.
March 23rd 2009, 06:15 PM
Pinkk

Chain rule (version 1)

If $z=f(x,y)$ and $x=x(t),y=y(t)$ then we can have the following:

$\frac{dz}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$

$\frac{\partial f}{\partial x}=ycos(xy)$
$\frac{\partial f}{\partial y}=xcos(xy)$

Since we are not given what $x(t),y(t)$ are equal to in terms of $t$ , we write the following

$\frac{dz}{dt}=ycos(xy)\frac{dx}{dt}+xcos(xy)\frac{ dy}{dt}$

For the second problem:

$\frac{d}{dx}(sin(x^{2})+cos^{2}x)=2xcos(x^{2})-2cosxsinx$ .

Remember, $\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$

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