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