To determine

To find: y"

Answer

y" = -5x4y11

Explanation

Formula:

i. Power rule:

ddxxn=nxn-1

ii. Difference rule:

ddxfx-gx=ddxfx-ddx(gx)

iii. Constant function rule:

ddxC=0

iv. Quotient rule:

ddxfxgx=gxddxfx-fxddx(gx)gx2

Given: x6+y6=1

Calculation:

Consider the function x6+y6=1

Differentiate with respect to x,

ddx(x6+y6)=ddx(1)

By using constant function and sum rule,

ddx(x6)+ddx(y6)=0

By using power rule,

6x5+6y5dydx=0

⇒6y5dydx=-6x5

⇒dydx=-x5y5

⇒y'=-x5y5

Again differentiate with respect to x,

y"=ddx-x5y5

By using quotient rule,

y" = y5ddx-x5--x5ddx(y5)y52

By using power rule,

y" = -5x4y5+5y4x5dydxy52

⇒y" = -5x4y5+5y4x5y'y10

Substitute, y'=-x5y5

y" = -5x4y5+5y4x5-x5y5y10

By solving it,

y" = -5x4y5-5x10yy10

By cross multiplication in numerator,

y" = -5x4y6-5x10yy10

⇒y" = -5x4y6-5x10y11

Taking -5x4 common,

y" = -5x4(y6+x6)y11

Since, x6+y6=1

y" = -5x4y11

Conclusion:

y" = -5x4y11