Question

Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. $$ \frac{\partial h}{\partial t_{2}} ; h(x, y)=\frac{x}{y}, x=t_{1} t_{2}, y=t_{1}^{2} t_{2} $$

Name: Problem 9, Chapter 14: Calculus: Early Transcendentals
Uploaded: 2021-08-16T05:21:42-08:00
Duration: 1 min 33 s
Channel: Linda Hand
Description: Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. (∂h)/(∂t2) ; h(x, y)=(x)/(y), x=t1 t2, y=t1^2 t2

   Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables.
$$
\frac{\partial h}{\partial t_{2}} ; h(x, y)=\frac{x}{y}, x=t_{1} t_{2}, y=t_{1}^{2} t_{2}
$$

Calculus: Early Transcendentals

Jon Rogawski, Colin… 4th Edition

Chapter 14, Problem 9

Instant Answer

Solved by Expert Linda Hand

08/16/2021

Step 1

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, $h$ is a function of $x$ and $y$, and $x$ and $y$ are functions of $t_1$ and $t_2$. Show more…

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Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. $$ \frac{\partial h}{\partial t_{2}} ; h(x, y)=\frac{x}{y}, x=t_{1} t_{2}, y=t_{1}^{2} t_{2} $$

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Key Concepts

Partial Differentiation

Partial differentiation is the process of finding the derivative of a function with respect to one variable while holding other variables constant. This concept is essential in multivariable calculus as it allows one to explore how changes in one particular independent variable affect the function, regardless of the influence of the other variables.

Multivariable Chain Rule

This concept involves calculating the derivative of a function that is composed of other functions, where each intermediary function depends on one or more variables. In a multivariable context, the chain rule states that to find the derivative of the outer function with respect to an independent variable, one sums the products of the partial derivatives of the outer function with respect to each intermediate variable and the derivative of that intermediate variable with respect to the independent variable.

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