   Chapter 3.5, Problem 13E

Chapter
Section
Textbook Problem

Find dy/dx by implicit differentiation.13. x + y = x 4 + y 4

To determine

To find: The derivative dydx by implicit differentiation.

Explanation

Given:

The equation x+y=x4+y4

Derivative rules: Chain rule

If y=f(u) and u=g(x)  are both differentiable function, then dydx=dydududx.

Calculation:

Obtain the derivative of x+y=x4+y4 implicit with respect to x.

x+y=x4+y4

Differentiate with respect to x on both sides,

ddx(x+y)=ddx(x4+y4)ddx((x+y)12)=ddx(x4)+ddx(y4)ddx((x+y)12)=4x3+ddx(y4)

Let u=x+y and apply the chain rule,

ddx((u)12)=4x3+ddx(y4)[ddu((u)12)dudx]=4x3+[ddy(y4)dydx]12u121dudx=4x3+4y3dydx12u12dudx=4x3+4y3dydx

Substitute the value u=x+y,

12(x+y)12ddx(x+y)=4x3+4y

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