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

[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)).

 

[g_edgar]

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

 

[HallsofIvy]

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

 

[SirTristan]

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

Thank you very much :)

 

[n!kofeyn]

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)).

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.