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 as constants.
Step 1: Substitute the expressions for x1 and x2 into the function y.
y = [((s+2)^2) + (s+2)(s^2 + t^2 + t) + ((s^2 + t^2 + t)^2)] / [(s+2) + (s^2 + t^2 + t)]
Step 2: Simplify the expression of y.
y = [(s^2 + 4s + 4) + (s^3 + s^2(t^2 + t) + 2s(t^2 + t) + (t^2 + t)^2)] / [s + 2 + s^2 + t^2 + t]
y = [s^2 + s^3 + 2s + 4 + s^2(t^2 + t) + 2s(t^2 + t) + (t^2 + t)^2] / [s^2 + s + 2 + t^2 + t]
Step 3: Differentiate y with respect to s using the quotient rule.
d/ds [y] = [denominator* d/ds(numerator) - numerator* d/ds(denominator)] / (denominator)^2
Let's find the partial derivative of the numerator and denominator separately.
a) Partial derivative of the numerator:
Differentiate each term of the numerator with respect to s, while treating t as a constant.
d/ds [s^2 + s^3 + 2s + 4 + s^2(t^2 + t) + 2s(t^2 + t) + (t^2 + t)^2]
= 2s + 3s^2 + 2 + 2s(t^2 + t) + 2(t^2 + t) + 0
= 3s^2 + 2s(t^2 + t) + 2s + 2(t^2 + t) + 2
= 3s^2 + 2s(t^2 + 1) + 2(t^2 + t) + 2
b) Partial derivative of the denominator:
Differentiate the denominator with respect to s, treating t as a constant.
d/ds [s^2 + s + 2 + t^2 + t]
= 2s + 1 + 0 + 0 + 0
= 2s + 1
Step 4: Evaluate the partial derivative of y with respect to s.
d/ds [y] = [ (2s + 1) * (3s^2 + 2s(t^2 + 1) + 2s + 2(t^2 + t) + 2) - (s^2 + s^3 + 2s + 4 + s^2(t^2 + t) + 2s(t^2 + t) + (t^2 + t)^2) * (2s + 1) ] / [ (s^2 + s + 2 + t^2 + t)^2 ]
Simplifying the expression further will provide the partial derivative of y with respect to s.