Partial derivatives of f(x)*(f(y)+f(z))?

1. Jul 11, 2009

SirTristan

Say you have something like f(x)*(f(y)+f(z)). What are the partial derivatives with respect to each variable? What rules are involved?

And how would this differ from f(x)*(g(x)+h(x)).

2. Jul 12, 2009

g_edgar

$$u = f(x)\;\big(f(y)+f(z)\big)$$
then
$$\frac{\partial u}{\partial x} = f'(x)\;\big(f(y)+f(z)\big)$$
$$\frac{\partial u}{\partial y} = f(x)\;\big(f'(y)\big)$$
$$\frac{\partial u}{\partial z} = f(x)\;\big(f'(z)\big)$$

3. Jul 12, 2009

HallsofIvy

Staff Emeritus
This has nothing that I can see to do with differential equations so I am moving it to Calculus and Analysis.

4. Jul 12, 2009

SirTristan

Thank you very much :)

5. Jul 13, 2009

n!kofeyn

The reasoning behind this is that when you take the partial derivative with respect to say x, you treat all the other variables, y and z, as constants. Then in that case f(y) and f(z) would be treated as constants. Then you take the derivatives with respect to each variable as normal.

It differs from f(x)*(g(x)+h(x)) because then all three functions are functions of the variable x. Then the partial derivatives with respect to y and z would be zero in this case.

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