# Partial derivatives problem

• Apr 21st 2010, 02:05 PM
john1991
Partial derivatives problem
1. a) Calculate the partial derivatives for \$\displaystyle f(x,y)=x^2 siny\$ and prove that

b)If \$\displaystyle x(u)=u^2\$ and \$\displaystyle y(u)=1/u\$, calculate \$\displaystyle df/du\$ using the chain rule.

c)For \$\displaystyle g(r)=r^4\$ where \$\displaystyle r=sqrt(x^2+y^2+z^2)\$ use the chain rule to calculate \$\displaystyle dg/dx\$, \$\displaystyle d^2g/dx2\$ and \$\displaystyle d^2g/dxdy\$.

I am having a lot of trouble with this problem.
• Apr 21st 2010, 02:58 PM
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Quote:

Originally Posted by john1991
1. a) Calculate the partial derivatives for \$\displaystyle f(x,y)=x^2 siny\$ and prove that

b)If \$\displaystyle x(u)=u^2\$ and \$\displaystyle y(u)=1/u\$, calculate \$\displaystyle df/du\$ using the chain rule.

c)For \$\displaystyle g(r)=r^4\$ where \$\displaystyle r=sqrt(x^2+y^2+z^2)\$ use the chain rule to calculate \$\displaystyle dg/dx\$, \$\displaystyle d^2g/dx2\$ and \$\displaystyle d^2g/dxdy\$.

I am having a lot of trouble with this problem.

a) To find the partial derivative of f(x,y) with respect to x, treat y as a constant, and vice versa. Have you tried this?

b) Maybe someone else can help me out here because I don't see how the chain rule applies. Replace all instances of x with x(u) = u^2 and likewise for y in f(x,y). You are left with a univariate problem of finding the derivative of f(u) with respect to u. (Here we are overloading the variable name f as in: f(u) = f(x(u),y(u)), which may be confusing, but that seems to be the notation of the problem.)

c) First of all, g(r) simplifies to \$\displaystyle g(x,y,z)=(x^2+y^2+z^2)^2\$. For the first partial derivative, you'll solve exactly as you would

d/dx (x^2 + c)^2

As in,

d/dx (x^2 + c)^2 = 2*(x^2 + c) * 2x

Hopefully it makes more sense now?