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.

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

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

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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

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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

Geometrical and Physical Optics

Electricity

Radiation

Kinematics Physics

Engineering Physics

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

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