Find the partial derivative y with respect to s for the following function:
y=[((x1)^2)+(x1)(x2)+((x2)^2)]/((x1)+(x2))
where x1=s+2 and x2=s^2+t^2+t .
x1 means x subscript 1
x2 means x subscript 2
To find the partial derivative of y with respect to s, we will differentiate each term of the function separately.
Step 1: Express y in terms of x1 and x2:
y = [(x1)^2 + x1*x2 + (x2)^2] / (x1 + x2)
Step 2: Find the derivatives of x1 and x2 with respect to s:
d(x1)/ds = 1
d(x2)/ds = 2s + dt/ds
Step 3: Differentiate each term of y with respect to s:
d/ds[(x1)^2] = 2x1 * d(x1)/ds
= 2(x1) * 1
= 2(x1)
d/ds[(x2)^2] = 2x2 * d(x2)/ds
= 2(x2) * (2s + dt/ds)
d/ds[x1*x2] = x2 * d(x1)/ds + x1 * d(x2)/ds
= x2 * 1 + x1 * (2s + dt/ds)
= x2 + 2sx1 + x1*dt/ds
d/ds[(x1 + x2)] = d(x1)/ds + d(x2)/ds
= 1 + (2s + dt/ds)
Step 4: Substitute the expressions for x1 and x2:
x1 = s + 2
x2 = s^2 + t^2 + t
d/ds[(x1)^2] = 2(x1) = 2(s + 2)
d/ds[(x2)^2] = 2(x2) * (2s + dt/ds) = 2(s^2 + t^2 + t) * (2s + dt/ds)
d/ds[x1*x2] = x2 + 2sx1 + x1*dt/ds = (s^2 + t^2 + t) + 2s(s + 2) + (s + 2)*dt/ds
d/ds[(x1 + x2)] = 1 + (2s + dt/ds) = 1 + 2s + dt/ds
Step 5: Substitute the expressions back into the original equation for y:
y = [(2(s + 2)) + ((s^2 + t^2 + t) + 2s(s + 2) + (s + 2)*dt/ds)] / (1 + 2s + dt/ds)
Simplifying the equation further is not possible since we don't have the expression for dt/ds. This is the partial derivative of y with respect to s for the given function.
To find the partial derivative of y with respect to s, we need to apply the chain rule and differentiate each term with respect to s.
Given:
y = [(x1)^2 + (x1)(x2) + (x2)^2] / (x1 + x2)
x1 = s + 2
x2 = s^2 + t^2 + t
Step 1: Rewrite the function using x1 and x2:
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: Differentiate each term with respect to s:
To differentiate (s + 2)^2, use the power rule for differentiation: d/ds (s^n) = n * (s^(n-1))
So, the derivative of (s + 2)^2 with respect to s is 2(s + 2) * 1 = 2(s + 2)
To differentiate (s + 2)(s^2 + t^2 + t), we can distribute the term:
= s(s^2 + t^2 + t) + 2(s^2 + t^2 + t)
= s^3 + st^2 + st + 2s^2 + 2t^2 + 2t
To differentiate (s^2 + t^2 + t)^2, we can use the chain rule:
Let u = s^2 + t^2 + t, then (s^2 + t^2 + t)^2 = u^2
Using the chain rule, the derivative is: d/ds (u^2) * du/ds
= 2u * (2(s^2 + t^2 + t))
= 4(s^2 + t^2 + t)(s^2 + t^2 + t)
To differentiate (s + 2) + (s^2 + t^2 + t), differentiate each term separately:
d/ds (s + 2) = 1
d/ds (s^2 + t^2 + t) = 2s + 1 (using the power rule for differentiation)
Step 3: Substitute the derivatives back into the original expression for y:
y = [2(s + 2) + (s^3 + st^2 + st + 2s^2 + 2t^2 + 2t) + 4(s^2 + t^2 + t)(s^2 + t^2 + t)] / [1 + 2s + 1 + 2s + 2t^2 + 2t]
Simplifying the expression will give you the partial derivative of y with respect to s.