Question

For the following exercises, calculate the partial derivatives. Let $z=\sinh (2 x+3 y) .$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$

Name: Problem 126, Chapter 4: Calculus Volume 3
Uploaded: 2019-08-07T20:27:05-08:00
Duration: 2 min 14 s
Channel: Bahar Tehranipoor
Description: For the following exercises, calculate the partial derivatives. Let z=sinh(2 x+3 y) . Find (∂z)/(∂x) and (∂z)/(∂y)

   For the following exercises, calculate the partial derivatives.
Let $z=\sinh (2 x+3 y) .$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$

Calculus Volume 3

Gilbert Strang 1st Edition

Chapter 4, Problem 126

Instant Answer

Solved by Expert Bahar Tehranipoor

08/07/2019

Step 1

We treat $y$ as a constant and differentiate $\sinh (2 x+3 y)$ with respect to $x$. The derivative of $\sinh u$ is $\cosh u$, so we get $\cosh (2 x+3 y)$. But we also need to multiply by the derivative of the inside function $2x+3y$ with respect to $x$, which is Show more…

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For the following exercises, calculate the partial derivatives. Let $z=\sinh (2 x+3 y) .$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$

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Key Concepts

Partial Derivatives

In multivariable calculus, partial derivatives measure how a function changes as one variable changes while keeping other variables constant. They are a fundamental tool for analyzing functions of several variables and understanding their local behavior.

Chain Rule

The chain rule is a method for differentiating composite functions. It allows us to compute the derivative of a function with respect to an inner function that depends on other variables by accounting for the rate of change of both the outer and inner functions.

Hyperbolic Functions

Hyperbolic functions, such as sinh and cosh, are analogs of the trigonometric functions but are defined using exponential functions. Their derivatives have specific formulations; for example, the derivative of sinh(u) with respect to u is cosh(u), which is used in the process of differentiation.

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