14. Compute the partial derivativeIn(x² + xy).

∂ ∂ xlnx2+xyTreat y as a constantApply chain Rule,∂ ∂ xlnx2+xy=1x2+xy∂ ∂ xx2+xy (…

Answered: Compute the partial derivative In(x² +…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter9: Multivariable Calculus

Section9.3: Maxima And Minima

Problem 20E

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Question

14. Compute the partial derivative
In(x² + xy).

Expert Solution

Step 1

$\frac{\partial}{\partial x} (\ln (x^{2} + x y)) Treat y as a constant Apply chain Rule, \frac{\partial}{\partial x} (\ln (x^{2} + xy)) = \frac{1}{x^{2} + xy} \frac{\partial}{\partial x} (x^{2} + xy) (\frac{\partial}{\partial x} (f (x)) = \frac{1}{f (x)} xf' (x)) \frac{\partial}{\partial x} (\ln (x^{2} + xy)) = \frac{1}{x^{2} + xy} (2 x + y) (\frac{\partial}{\partial x} (x^{2} + xy) = \frac{\partial}{\partial x} (x^{2}) + \frac{\partial}{\partial x} (xy) = 2 x + y) \frac{\partial}{\partial x} (\ln (x^{2} + xy)) = \frac{2 x + y}{x^{2} + xy}$