Answer :
Let's determine the partial derivatives of the composite function [tex]\[ f(x, y, z) = x + 8y^2 - z^2 \][/tex] where [tex]\(x = ut\)[/tex], [tex]\(y = e^{u + 9v + 2w + 8t}\)[/tex], and [tex]\(z = u + \frac{1}{2}v + 6t\)[/tex].
Step 1: Express [tex]\(f\)[/tex] in terms of [tex]\(u, v, w,\)[/tex] and [tex]\(t\)[/tex].
Recall the function definition:
[tex]\[ f(u, v, w, t) = x + 8y^2 - z^2 \][/tex]
Substitute the expressions for [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]:
[tex]\[ x = ut \][/tex]
[tex]\[ y = e^{u + 9v + 2w + 8t} \][/tex]
[tex]\[ z = u + \frac{1}{2}v + 6t \][/tex]
Thus, we have:
[tex]\[ f(u, v, w, t) = ut + 8(e^{u + 9v + 2w + 8t})^2 - (u + \frac{1}{2}v + 6t)^2 \][/tex]
Step 2: Calculate [tex]\(\frac{\partial f}{\partial u}\)[/tex].
[tex]\[ f = ut + 8(e^{u + 9v + 2w + 8t})^2 - (u + \frac{1}{2}v + 6t)^2 \][/tex]
[tex]\[ \frac{\partial f}{\partial u} = \frac{\partial}{\partial u} \left( ut \right) + \frac{\partial}{\partial u} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) - \frac{\partial}{\partial u} \left( (u + \frac{1}{2}v + 6t)^2 \right) \][/tex]
[tex]\[ \frac{\partial}{\partial u}(ut) = t \][/tex]
[tex]\[ \frac{\partial}{\partial u} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) = 16(e^{u + 9v + 2w + 8t}) \cdot e^{u + 9v + 2w + 8t} \cdot 1 = 16 e^{2(u + 9v + 2w + 8t)} \][/tex]
[tex]\[ \frac{\partial}{\partial u} \left( (u + \frac{1}{2}v + 6t)^2 \right) = 2(u + \frac{1}{2}v + 6t) \][/tex]
Summarizing:
[tex]\[ \frac{\partial f}{\partial u} = t + 16 e^{2(u + 9v + 2w + 8t)} - 2(u + \frac{1}{2}v + 6t) \][/tex]
Combining the terms:
[tex]\[ \frac{\partial f}{\partial u} = 16 e^{2(u + 9v + 2w + 8t)} - 11t - 2u - v/2 \][/tex]
Step 3: Calculate [tex]\(\frac{\partial f}{\partial v}\)[/tex].
[tex]\[ \frac{\partial f}{\partial v} = \frac{\partial}{\partial v} \left( ut \right) + \frac{\partial}{\partial v} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) - \frac{\partial}{\partial v} \left( (u + \frac{1}{2}v + 6t)^2 \right) \][/tex]
[tex]\[ \frac{\partial}{\partial v}(ut) = 0 \][/tex]
[tex]\[ \frac{\partial}{\partial v} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) = 144 e^{2(u + 9v + 2w + 8t)} \][/tex]
[tex]\[ \frac{\partial}{\partial v} \left( (u + \frac{1}{2}v + 6t)^2 \right) = (u + \frac{1}{2}v + 6t) \][/tex]
Summarizing:
[tex]\[ \frac{\partial f}{\partial v} = 144 e^{2(u + 9v + 2w + 8t)} - u - \frac{1}{2} v - 6t \][/tex]
Step 4: Calculate [tex]\(\frac{\partial f}{\partial w}\)[/tex].
[tex]\[ \frac{\partial f}{\partial w} = \frac{\partial}{\partial w} \left( ut \right) + \frac{\partial}{\partial w} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) - \frac{\partial}{\partial w} \left( (u + \frac{1}{2}v + 6t)^2 \right) \][/tex]
[tex]\[ \frac{\partial}{\partial w}(ut) = 0 \][/tex]
[tex]\[ \frac{\partial}{\partial w} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) = 32 e^{2(u + 9v + 2w + 8t)} \][/tex]
[tex]\[ \frac{\partial}{\partial w} \left( (u + \frac{1}{2}v + 6t)^2 \right) = 0 \][/tex]
Summarizing:
[tex]\[ \frac{\partial f}{\partial w} = 32 e^{2(u + 9v + 2w + 8t)} \][/tex]
Step 5: Calculate [tex]\(\frac{\partial f}{\partial t}\)[/tex].
[tex]\[ \frac{\partial f}{\partial t} = \frac{\partial}{\partial t} \left( ut \right) + \frac{\partial}{\partial t} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) - \frac{\partial}{\partial t} \left( (u + \frac{1}{2}v + 6t)^2 \right) \][/tex]
[tex]\[ \frac{\partial}{\partial t}(ut) = u \][/tex]
[tex]\[ \frac{\partial}{\partial t} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) = 128 e^{2(u + 9v + 2w + 8t)} \][/tex]
[tex]\[ \frac{\partial}{\partial t} \left( (u + \frac{1}{2}v + 6t)^2 \right) = 72 (u + \frac{1}{2}v + 6t) \][/tex]
Summarizing:
[tex]\[ \frac{\partial f}{\partial t} = 128 e^{2(u + 9v + 2w + 8t)} - 11u - 6v - 72t \][/tex]
Thus, the partial derivatives are:
[tex]\[ \frac{\partial f}{\partial u} = 16 e^{2(u + 9v + 2w + 8t)} - 11t - 2u - v/2 \][/tex]
[tex]\[ \frac{\partial f}{\partial v} = 144 e^{2(u + 9v + 2w + 8t)} - u - \frac{1}{2} v - 6t \][/tex]
[tex]\[ \frac{\partial f}{\partial w} = 32 e^{2(u + 9v + 2w + 8t)} \][/tex]
[tex]\[ \frac{\partial f}{\partial t} = 128 e^{2(u + 9v + 2w + 8t)} - 11u - 6v - 72t \][/tex]
Step 1: Express [tex]\(f\)[/tex] in terms of [tex]\(u, v, w,\)[/tex] and [tex]\(t\)[/tex].
Recall the function definition:
[tex]\[ f(u, v, w, t) = x + 8y^2 - z^2 \][/tex]
Substitute the expressions for [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]:
[tex]\[ x = ut \][/tex]
[tex]\[ y = e^{u + 9v + 2w + 8t} \][/tex]
[tex]\[ z = u + \frac{1}{2}v + 6t \][/tex]
Thus, we have:
[tex]\[ f(u, v, w, t) = ut + 8(e^{u + 9v + 2w + 8t})^2 - (u + \frac{1}{2}v + 6t)^2 \][/tex]
Step 2: Calculate [tex]\(\frac{\partial f}{\partial u}\)[/tex].
[tex]\[ f = ut + 8(e^{u + 9v + 2w + 8t})^2 - (u + \frac{1}{2}v + 6t)^2 \][/tex]
[tex]\[ \frac{\partial f}{\partial u} = \frac{\partial}{\partial u} \left( ut \right) + \frac{\partial}{\partial u} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) - \frac{\partial}{\partial u} \left( (u + \frac{1}{2}v + 6t)^2 \right) \][/tex]
[tex]\[ \frac{\partial}{\partial u}(ut) = t \][/tex]
[tex]\[ \frac{\partial}{\partial u} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) = 16(e^{u + 9v + 2w + 8t}) \cdot e^{u + 9v + 2w + 8t} \cdot 1 = 16 e^{2(u + 9v + 2w + 8t)} \][/tex]
[tex]\[ \frac{\partial}{\partial u} \left( (u + \frac{1}{2}v + 6t)^2 \right) = 2(u + \frac{1}{2}v + 6t) \][/tex]
Summarizing:
[tex]\[ \frac{\partial f}{\partial u} = t + 16 e^{2(u + 9v + 2w + 8t)} - 2(u + \frac{1}{2}v + 6t) \][/tex]
Combining the terms:
[tex]\[ \frac{\partial f}{\partial u} = 16 e^{2(u + 9v + 2w + 8t)} - 11t - 2u - v/2 \][/tex]
Step 3: Calculate [tex]\(\frac{\partial f}{\partial v}\)[/tex].
[tex]\[ \frac{\partial f}{\partial v} = \frac{\partial}{\partial v} \left( ut \right) + \frac{\partial}{\partial v} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) - \frac{\partial}{\partial v} \left( (u + \frac{1}{2}v + 6t)^2 \right) \][/tex]
[tex]\[ \frac{\partial}{\partial v}(ut) = 0 \][/tex]
[tex]\[ \frac{\partial}{\partial v} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) = 144 e^{2(u + 9v + 2w + 8t)} \][/tex]
[tex]\[ \frac{\partial}{\partial v} \left( (u + \frac{1}{2}v + 6t)^2 \right) = (u + \frac{1}{2}v + 6t) \][/tex]
Summarizing:
[tex]\[ \frac{\partial f}{\partial v} = 144 e^{2(u + 9v + 2w + 8t)} - u - \frac{1}{2} v - 6t \][/tex]
Step 4: Calculate [tex]\(\frac{\partial f}{\partial w}\)[/tex].
[tex]\[ \frac{\partial f}{\partial w} = \frac{\partial}{\partial w} \left( ut \right) + \frac{\partial}{\partial w} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) - \frac{\partial}{\partial w} \left( (u + \frac{1}{2}v + 6t)^2 \right) \][/tex]
[tex]\[ \frac{\partial}{\partial w}(ut) = 0 \][/tex]
[tex]\[ \frac{\partial}{\partial w} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) = 32 e^{2(u + 9v + 2w + 8t)} \][/tex]
[tex]\[ \frac{\partial}{\partial w} \left( (u + \frac{1}{2}v + 6t)^2 \right) = 0 \][/tex]
Summarizing:
[tex]\[ \frac{\partial f}{\partial w} = 32 e^{2(u + 9v + 2w + 8t)} \][/tex]
Step 5: Calculate [tex]\(\frac{\partial f}{\partial t}\)[/tex].
[tex]\[ \frac{\partial f}{\partial t} = \frac{\partial}{\partial t} \left( ut \right) + \frac{\partial}{\partial t} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) - \frac{\partial}{\partial t} \left( (u + \frac{1}{2}v + 6t)^2 \right) \][/tex]
[tex]\[ \frac{\partial}{\partial t}(ut) = u \][/tex]
[tex]\[ \frac{\partial}{\partial t} \left( 8(e^{u + 9v + 2w + 8t})^2 \right) = 128 e^{2(u + 9v + 2w + 8t)} \][/tex]
[tex]\[ \frac{\partial}{\partial t} \left( (u + \frac{1}{2}v + 6t)^2 \right) = 72 (u + \frac{1}{2}v + 6t) \][/tex]
Summarizing:
[tex]\[ \frac{\partial f}{\partial t} = 128 e^{2(u + 9v + 2w + 8t)} - 11u - 6v - 72t \][/tex]
Thus, the partial derivatives are:
[tex]\[ \frac{\partial f}{\partial u} = 16 e^{2(u + 9v + 2w + 8t)} - 11t - 2u - v/2 \][/tex]
[tex]\[ \frac{\partial f}{\partial v} = 144 e^{2(u + 9v + 2w + 8t)} - u - \frac{1}{2} v - 6t \][/tex]
[tex]\[ \frac{\partial f}{\partial w} = 32 e^{2(u + 9v + 2w + 8t)} \][/tex]
[tex]\[ \frac{\partial f}{\partial t} = 128 e^{2(u + 9v + 2w + 8t)} - 11u - 6v - 72t \][/tex]
