Find the indicated partial derivative. (Assume a, b, and c are greater than three.)
u = x^(a)y^(b)z^(c)
[∂^(6)u]/[∂x ∂y^(2)∂z^(3)]=
U = x^a y^b z^c
Ux = ax^(a-1) * y^b * z^c
Uxy = ax^(a-1) * by^(b-1) * z^c
Uxyy = ax^(a-1) * b(b-1)y^(b-2) * z^c
See where this is going?
Well, well, well, look who's getting fancy with their partial derivatives! Alright, let's dive in!
We have this function u = x^(a)y^(b)z^(c), and we're looking to find the sixth partial derivative with respect to x, y squared, and z cubed. My oh my, it's going to get complicated!
To find the sixth partial derivative, we need to differentiate six times with respect to x, once with respect to y, twice with respect to y, and thrice with respect to z. Phew, that's going to be a wild ride!
Alright, let's break it down step by step. First, let's take the derivative of u with respect to x:
∂u/∂x = a x^(a-1) y^b z^c
Good job so far! Now, let's take the second derivative with respect to y:
∂^2u/∂y^2 = a (a-1) x^(a-1) y^(b-2) z^c
Alright, almost halfway there! Now, let's take the third derivative with respect to z:
∂^3u/∂z^3 = a (a-1) x^(a-1) y^(b-2) z^(c-3)
Just a few more to go! Now, we need to differentiate three more times with respect to x:
∂^6u/∂x ∂y^2 ∂z^3 = a (a-1)(a-2)(a-3)(a-4)(a-5) x^(a-6) y^(b-2) z^(c-3)
Bravo! You made it through the wilderness of partial derivatives! Now just plug in the values for a, b, and c, which should be greater than three, and you're golden!
I hope you enjoyed this rollercoaster ride of partial derivatives. If you have any more math conundrums or if you need a clown joke to lighten the mood, just let me know!
To find the indicated partial derivative, we need to differentiate the function u = x^(a)y^(b)z^(c) with respect to x, y, and z as indicated and take the 6th order derivative.
Step 1: Take the partial derivative with respect to x:
∂u/∂x = a*x^(a-1)*y^(b)*z^(c)
Step 2: Take the partial derivative with respect to y twice:
∂^(2)u/∂y^(2) = ∂/∂y(∂u/∂y) = ∂/∂y(b*y^(b-1)*z^(c))
= b*(b-1)*y^(b-2)*z^(c)
Step 3: Take the partial derivative with respect to z three times:
∂^(3)u/∂z^(3) = ∂/∂z(∂^(2)u/∂z^(2)) = ∂/∂z(b*(b-1)*y^(b-2)*z^(c))
= b*(b-1)*∂/∂z(y^(b-2)*z^(c))
= b*(b-1)*y^(b-2)*c*(c-1)*z^(c-2)
Step 4: Take the 6th order mixed partial derivative:
[∂^(6)u]/[∂x ∂y^(2)∂z^(3)] = ∂^(3)/∂z^(3)(∂^(2)/∂y^(2)(∂u/∂x))
= ∂^(3)/∂z^(3)(b*(b-1)*y^(b-2)*c*(c-1)*z^(c-2))
= b*(b-1)*c*(c-1)*(c-2)*y^(b-2)*z^(c-5)
Therefore, [∂^(6)u]/[∂x ∂y^(2)∂z^(3)] = b*(b-1)*c*(c-1)*(c-2)*y^(b-2)*z^(c-5)
To find the indicated partial derivative, we will first find the partial derivatives individually and then evaluate them.
Given function: u = x^a * y^b * z^c
Let's find the partial derivatives step by step:
1. ∂u/∂x: To find the partial derivative with respect to x, we treat y and z as constants and differentiate x^a with respect to x:
∂u/∂x = a * x^(a-1) * y^b * z^c
2. ∂²u/∂y²: To find the second partial derivative with respect to y, we need to differentiate ∂u/∂y with respect to y once again. Thus, we treat x and z as constants and differentiate y^b with respect to y:
∂²u/∂y² = a * (a - 1) * x^a * y^(b - 2) * z^c
3. ∂³u/∂z³: To find the third partial derivative with respect to z, we need to differentiate ∂²u/∂z² with respect to z once again. Thus, we treat x and y as constants and differentiate z^c with respect to z:
∂³u/∂z³ = a * (a - 1) * x^a * y^b * z^(c - 3)
Now that we have the individual partial derivatives, we can combine them to find the indicated partial derivative:
[∂⁶u]/[∂x ∂y²∂z³] = (∂³/∂z³)(∂²/∂y²)(∂/∂x)(x^a * y^b * z^c)
Substituting the partial derivatives we found earlier:
[∂⁶u]/[∂x ∂y²∂z³] = (a * (a - 1) * x^a * y^b * z^(c - 3))(a * (a - 1) * x^a * y^(b - 2) * z^c)(a * x^(a - 1) * y^b * z^c)
After simplifying, the final expression for the indicated partial derivative would be:
[∂⁶u]/[∂x ∂y²∂z³] = a^3 * (a - 1)^3 * x^(3a - 3) * y^(3b - 3) * z^(3c - 9)
Therefore, the indicated partial derivative [∂⁶u]/[∂x ∂y²∂z³] of the function u = x^a * y^b * z^c is a^3 * (a - 1)^3 * x^(3a - 3) * y^(3b - 3) * z^(3c - 9).