Question

Question #dc166

Answer 1

`f_a(a,b) = a/sqrt(a^2+b^2)`

`f_b(a,b) = b/sqrt(a^2+b^2)`

Explanation:

When taking the derivative of a function with respect to a given variable, we treat all other variables as constants. Then, for each of the given problems, we can find the derivatives using the chain rule and the power rule .

`f_a(a,b) = d/(da)sqrt(a^2+b^2)`

`=d/(da)(a^2+b^2)^(1/2)`

`=1/2(a^2+b^2)^(1/2-1)(d/(da)(a^2+b^2))`

`=1/2(a^2+b^2)^(-1/2)((d/(da)a^2)+(d/(da)b^2))`

`=1/(2(a^2+b^2)^(1/2))(2a^1+0)`

`=a/sqrt(a^2+b^2)`

We could also go through the same process to find `f_b(a,b)`, however because the equation is completely symmetrical in `a` and `b`, we can simply replace `a` with `b` and `b` with `a` in the prior derivative to get

`f_b(a,b) = b/sqrt(a^2+b^2)`