Find the partial derivative y with respect to s for the following function:
y=[((x_1)^2)+(x_1)(x_2)+((x_2)^2)]/((x_1)+(x_2))
where x_1=s+2 and x_2=s^2+t^2+t .
The underscore (_) stands for subscript
To find the partial derivative of y with respect to s, we need to differentiate the function y with respect to s while treating all other variables (in this case, t) as constants.
Let's start by substituting the expressions for x_1 and x_2 into the function y:
y = [(x_1^2) + x_1x_2 + (x_2^2)] / (x_1 + x_2)
= [((s+2)^2) + (s+2)(s^2+t^2+t) + (s^2+t^2+t)^2] / ((s+2) + (s^2+t^2+t))
Next, we'll simplify the expression by expanding it:
y = [(s^2 + 4s + 4) + (s^3 + 2s^2 + st^2 + st + 2s^2 + 4st + 4t^2 + 2t) + (s^4 + 2s^2t^2 + 2s^2t + s^2 + 2st^3 + 2st^2 + 2st + 4t^4 + 4t^3 + 2t^2)] / (s + 2 + s^2 + t^2 + t)
y = [s^4 + 3s^3 + 7s^2 + 12st + 8t^2 + 2st^2 + 2st + 4t^3 + 4t^4] / (s^2 + s + t^2 + 2t + 2)
Now, let's differentiate y with respect to s. We treat all other variables (t) as constants and differentiate each term of y one by one:
∂y/∂s = [(4s^3 + 9s^2 + 14st + 12t + 0 + 2t + 2t) * (s^2 + s + t^2 + 2t + 2) - (s^4 + 3s^3 + 7s^2 + 12st + 8t^2 + 2st^2 + 2st + 4t^3 + 4t^4) * (2s + 1)] / (s^2 + s + t^2 + 2t + 2)^2
Now, simplify the expression:
∂y/∂s = (4s^5 + 9s^4 + 14s^2t + 12st + 8t^2s^2 + 2t^3 + 4s^3 + 2ts^2 + 2st^2 + 14s^2 + 2t^3 + 8t^2 + 16t - 2s^5 - 3s^4 - 7s^3 - 16s^2t - 8t^2s^2 - 2t^3 - 2t^3 - 2st^2 - 4t^2s - 4t^3) / (s^4 + 2s^3 + 3s^2 + 2s + t^2 + 2t)^2
Finally, simplify the expression further if needed.
The partial derivative of y with respect to s, denoted ∂y/∂s, is given by the expression above.