Chapter 1: Q6P (page 1)

Given , $z = {(x + y)}^{5}, y = \sin 10 x, find \frac{dz}{dx} .$ .

Short Answer

Expert verified

The $\frac{d z}{d x} is 5 {(x + y)}^{4} (1 + 10 \cos 10 x)$ .

Step by step solution

Explanation of solution

The provided expressions are, $z = {(x + y)}^{5}, y = \sin 10 x$ .

Chain Rule

The number of functions in the composition affects how many differentiation steps are required when using the chain rule to get the derivative of composite functions.

Calculation

Complete differentiation is expressed using partial derivatives.

$\frac{d z}{d x} = \frac{鈭� z}{鈭� x} d x + \frac{鈭� z}{鈭� y} \frac{d y}{d x} - - - - (1)$

$Find \frac{鈭� z}{鈭� x} . \frac{鈭� z}{鈭� x} = \frac{鈭� {(x + y)}^{5}}{鈭� x} = 5 {(x + y)}^{4} \frac{鈭� {(x + y)}^{5}}{鈭� x} = 5 (x + y) 4 Find \frac{鈭� z}{鈭� y} . \frac{鈭� z}{鈭� y} = \frac{鈭� {(x + y)}^{5}}{鈭� y} = 5 {(x + y)}^{4} \frac{鈭� {(x + y)}^{5}}{鈭� y} = 5 (x + y) 4 Find \frac{dy}{dx} . \frac{dy}{dx} = \frac{d (\sin 10 x)}{dx} = 10 \cos 10 x Find \frac{dy}{dt} . \frac{dy}{dt} = \frac{d (\cos t)}{dt} = - \sin t find \frac{dz}{dx} use equation (1) and above results \frac{dz}{dx} = 5 {(x + y)}^{4} + 5 {(x + y)}^{4} (10 \cos 10 x) = 5 {(x + y)}^{4} (1 + 10 \cos 10 x)$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Given , $z = {(x + y)}^{5}, y = \sin 10 x, find \frac{dz}{dx} .$ .

Short Answer

Step by step solution

Explanation of solution

Chain Rule

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Electromagnetism

Fundamentals of Physics

Electrostatics

Fields in Physics

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

91影视

Given ,z=(x+y)5,y=sin10x,finddzdx..

Short Answer

Step by step solution

Explanation of solution

Chain Rule

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Electromagnetism

Fundamentals of Physics

Electrostatics

Fields in Physics

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

Given , $z = {(x + y)}^{5}, y = \sin 10 x, find \frac{dz}{dx} .$ .