Given the function (a) Determine ƒ₂ (b) Determine ƒ y (c) Determine f yyz f(x, y, z) = (sin yz)e²³-¹√√
Given the function (a) Determine ƒ₂ (b) Determine ƒ y (c) Determine f yyz f(x, y, z) = (sin yz)e²³-¹√√
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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![### Calculating Partial Derivatives of the Given Function
Given the function:
\[ f(x, y, z) = (\sin yz) e^{z^3 - z^{-1}} \sqrt{y} \]
We need to determine the following partial derivatives:
#### (a) Determine \( f_z \)
To find the partial derivative of \( f \) with respect to \( z \), denoted as \( f_z \), we apply the chain rule to each term that contains \( z \).
#### (b) Determine \( f_y \)
To find the partial derivative of \( f \) with respect to \( y \), denoted as \( f_y \), we apply the chain rule to each term that contains \( y \).
#### (c) Determine \( f_{yyz} \)
To find the mixed partial derivative \( f_{yyz} \), we first find \( f_y \), then take the second partial derivative with respect to \( y \), and finally take the partial derivative with respect to \( z \). For clarity, step-by-step derivations need to be written out.
The computation involves applying product and chain rules multiple times to differentiate the composite function provided above.
### Notes
1. Use the following differentiation rules during calculations:
- Product rule: \((uv)' = u'v + uv'\)
- Chain rule: \(\frac{d}{dx}g(h(x)) = g'(h(x))h'(x)\)
2. For an exponential function \( e^{u} \), where \( u \) is a function of multiple variables, apply the chain rule.
Breaking the function into its component parts helps in managing the differentiation process. Ensure to handle the trigonometric function \(\sin(yz)\), the exponential function \(e^{z^3 - z^{-1}}\), and the square root function \(\sqrt{y}\) meticulously. Each term needs to be differentiated concerning the specified variable while treating others as constants.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa40558bc-afda-4e2b-be1a-c4211c27f73c%2F26b27ef0-811e-4391-96b6-127381c56dfe%2Fobs249l_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating Partial Derivatives of the Given Function
Given the function:
\[ f(x, y, z) = (\sin yz) e^{z^3 - z^{-1}} \sqrt{y} \]
We need to determine the following partial derivatives:
#### (a) Determine \( f_z \)
To find the partial derivative of \( f \) with respect to \( z \), denoted as \( f_z \), we apply the chain rule to each term that contains \( z \).
#### (b) Determine \( f_y \)
To find the partial derivative of \( f \) with respect to \( y \), denoted as \( f_y \), we apply the chain rule to each term that contains \( y \).
#### (c) Determine \( f_{yyz} \)
To find the mixed partial derivative \( f_{yyz} \), we first find \( f_y \), then take the second partial derivative with respect to \( y \), and finally take the partial derivative with respect to \( z \). For clarity, step-by-step derivations need to be written out.
The computation involves applying product and chain rules multiple times to differentiate the composite function provided above.
### Notes
1. Use the following differentiation rules during calculations:
- Product rule: \((uv)' = u'v + uv'\)
- Chain rule: \(\frac{d}{dx}g(h(x)) = g'(h(x))h'(x)\)
2. For an exponential function \( e^{u} \), where \( u \) is a function of multiple variables, apply the chain rule.
Breaking the function into its component parts helps in managing the differentiation process. Ensure to handle the trigonometric function \(\sin(yz)\), the exponential function \(e^{z^3 - z^{-1}}\), and the square root function \(\sqrt{y}\) meticulously. Each term needs to be differentiated concerning the specified variable while treating others as constants.
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