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Chapter 21: Problem 8

$$ \text { If } V=D^{1 / 4} T^{-5 / 6} \text { find } \frac{\partial V}{\partial T} \text { and } \frac{\partial V}{\partial D} $$

Short Answer

Expert verified
Question: Find the partial derivatives of the function, V = D^(1/4) * T^(-5/6), with respect to T and D. Answer: The partial derivative of V with respect to T is: ∂V/∂T = -5/6 D^(1/4) T^(-11/6). The partial derivative of V with respect to D is: ∂V/∂D = 1/4 D^(-3/4) T^(-5/6).

Step by step solution

01

Understand the function

We are given the function: $$ V = D^{\frac{1}{4}}T^{-\frac{5}{6}} $$
02

Find the partial derivative with respect to T

To find the partial derivative of V with respect to T, we will treat D as a constant. So, applying the power rule of differentiation, we get: $$ \frac{\partial V}{\partial T} = D^{\frac{1}{4}} \cdot (-\frac{5}{6}) T^{-\frac{5}{6} - 1} $$ which simplifies to: $$ \frac{\partial V}{\partial T} = -\frac{5}{6} D^{\frac{1}{4}} T^{-\frac{11}{6}} $$
03

Find the partial derivative with respect to D

To find the partial derivative of V with respect to D, we will treat T as a constant. So, applying the power rule of differentiation, we get: $$ \frac{\partial V}{\partial D} = \frac{1}{4} D^{\frac{1}{4} - 1} T^{-\frac{5}{6}} $$ which simplifies to: $$ \frac{\partial V}{\partial D} = \frac{1}{4} D^{-\frac{3}{4}} T^{-\frac{5}{6}} $$ As a result, we have found the required partial derivatives: $$ \frac{\partial V}{\partial T} = -\frac{5}{6} D^{\frac{1}{4}} T^{-\frac{11}{6}} $$ and $$ \frac{\partial V}{\partial D} = \frac{1}{4} D^{-\frac{3}{4}} T^{-\frac{5}{6}} $$

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