Let f(x, y) = (x³ + ay², x²y°, 2x + by), g(u, v, w) = (3u² + bv² + w*, uvw).Find the derivative of f(g(u, v, w)) using the chain rule.

Name: Let f(x, y) = (x³ + ay², x²y°, 2x + by), g(u, v, w) = (3u² + bv² + w*
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Description: Let f(x, y) = (x³ + ay², x²y°, 2x + by), g(u, v, w) = (3u² + bv² + w*, uvw).Find the derivative of f(g(u, v, w)) using the chain rule.

Answered: Let f(x, y) = (x° + ay´, xʻy°, 2x+by),…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let f(x, y) = (x³ + ay², x²y°, 2x + by), g(u, v, w) = (3u² + bv² + w*, uvw).
Find the derivative of f(g(u, v, w)) using the chain rule.

Expert Solution

Given functions f and g such that

$f (x, y) = (x^{3} + {ay}^{2}, x^{2} y^{3}, 2 x + by) g (u, v, w) = (3 u^{2} + {bv}^{2} + w^{3}, uvw) To Find : Derivative of f (g (u, v, w)) using the chain rule .$

Note:

$We use the chain rule to differentiate composite function : If f and g are both differentiable and F = f ° g (i . e . F (x) = f (g (x))) is the composite function, then F' (x) = f' (g (x)) ∙ g' (x) Now, derivative of a multivariable vector function is given as : \frac{\partial F}{\partial x_{i}} = ⟨ \frac{\partial F_{1}}{\partial x_{i}}, \frac{\partial F_{2}}{\partial x_{i}}, \dots, \frac{\partial F_{m}}{\partial x_{i}} ⟩$

We begin with finding the composite function.

$F = f \circ g = f (g (u, v, w)) = f ((3 u^{2} + {bv}^{2} + w^{3}, uvw)) = ((3 u^{2} + {bv}^{2} + w^{3})^{3} + a (uvw)^{2}, (3 u^{2} + {bv}^{2} + w^{3})^{2} (uvw)^{3}, 2 (3 u^{2} + {bv}^{2} + w^{3}) + b (uvw))$

$Next, we find the derivative of the f (g (u, v, w)) using the chain rule . Let F_{1} = (3 u^{2} + {bv}^{2} + w^{3})^{3} + a (uvw)^{2}, F_{2} = (3 u^{2} + {bv}^{2} + w^{3})^{2} (uvw)^{3}, F_{3} = 2 (3 u^{2} + {bv}^{2} + w^{3}) + b (uvw) \frac{\partial F}{\partial u} = ⟨ \frac{\partial F_{1}}{\partial u}, \frac{\partial F_{2}}{\partial u}, \frac{\partial F_{3}}{\partial u} ⟩ \frac{\partial F}{\partial v} = ⟨ \frac{\partial F_{1}}{\partial v}, \frac{\partial F_{2}}{\partial v}, \frac{\partial F_{3}}{\partial v} ⟩ \frac{\partial F}{\partial w} = ⟨ \frac{\partial F_{1}}{\partial w}, \frac{\partial F_{2}}{\partial w}, \frac{\partial F_{3}}{\partial w} ⟩$

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