To determine

To find: The derivative dydx by implicit differentiation.

Answer

The differentiation of x+y=x4+y4 is dydx=8x3(x+y)12−11−8y3(x+y)12.

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=dydu⋅dudx.

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]12u12−1dudx=4x3+4y3dydx12u−12dudx=4x3+4y3dydx

Substitute the value u=x+y,

12(x+y)−12ddx(x+y)=4x3+4y3dydx12(x+y)−12[ddx(x)+dydx]=4x3+4y3dydx(x+y)−12[1+dydx]=2(4x3+4y3dydx)(x+y)−12+dydx(x+y)−12=8x3+8y3dydx

Isolate the derivative dydx to one side of the equation,

dydx(x+y)−12−8y3dydx=8x3−(x+y)−12dydx(1(x+y)12−8y3)=8x3−1(x+y)12dydx(1−8y3(x+y)12)=8x3(x+y)12−1dydx=8x3(x+y)12−11−8y3(x+y)12

Therefore, the differentiation of x+y=x4+y4 is dydx=8x3(x+y)12−11−8y3(x+y)12.