Partial Differentiation

• Jan 10th 2009, 07:49 AM
KayPee
Partial Differentiation
Consider the function f(x,y)=x^2+y^2 where x=sin2θ and y=cos2θ

I know this involves the chain rule and trig id.

But I'm still not getting it

Kindly assist.
• Jan 10th 2009, 08:00 AM
Jester
Quote:

Originally Posted by KayPee
Consider the function f(x,y)=x^2+y^2 where x=sin2θ and y=cos2θ

I know this involves the chain rule and trig id.

But I'm still not getting it

Kindly assist.

Since $f$ is a function of $x \; \text{and}\; y$ and $x \; \text{and}\; y$ are functions of $\theta$, then $f$ is a function of $\theta$ so $\frac{df}{d \theta}$ makes sense. The chain rule is

$(1)\;\;\;\;\;\frac{df}{d \theta} = \frac{\partial f}{\partial x} \frac{d x}{d \theta} + \frac{\partial f}{\partial y} \frac{dy} {d \theta}$.

So calculate all of the derivatives

$\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{d x}{ d \theta} \; \text{and}\;\frac{d y}{d \theta}$

and substitute them into (1). Finally, use your expressions for $x\; \text{and}\; y$ to get your final answer.
• Jan 10th 2009, 08:10 AM
Prove It
Quote:

Originally Posted by KayPee
Consider the function f(x,y)=x^2+y^2 where x=sin2θ and y=cos2θ

I know this involves the chain rule and trig id.

But I'm still not getting it

Kindly assist.

What does $\sin^2{2\theta} + \cos^2{2\theta}$ equal?

What is the derivative of a constant?
• Jan 10th 2009, 09:55 AM
KayPee
Quote:

Originally Posted by Prove It
What does $\sin^2{2\theta} + \cos^2{2\theta}$ equal? one

What is the derivative of a constant?

0

how would these two answers help me arrive at the final answer

thanks
• Jan 10th 2009, 01:36 PM
mr fantastic
Quote:

Originally Posted by KayPee
0

how would these two answers help me arrive at the final answer

thanks

Look at your expression for f(x, y). Now look at what Prove It wrote. Join the dots.