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Find (dy)/(dx), when: (cosx)^(y)=(cosy...

Find `(dy)/(dx)`, when:
`(cosx)^(y)=(cosy)^(x)`

Text Solution

Verified by Experts

The correct Answer is:
`((logcosy+y tan x))/((log cos x+x tan y))`
`(cosx)^(y)=(cosy)^(x)`
`rArr ylog cos x= x log cos yy`
`rArry.((-sinx)/(cosx))+(logcos x)(dy)/(dx)=x.((-siny)/(cosy)).(dy)/(dx)+(log cosy).1`
`rArr(log cos x+x tany)(dy)/(dx)=(log cos y+y tan x).`
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RS AGGARWAL-DIFFERENTIATION-Exercise 10F
  1. Find (dy)/(dx), when: y=x^(x).e^((2x+5))

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  2. Find (dy)/(dx), when: y=(2x+3)^(5)(3x-5)^(7)(5x-1)^(3)

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  3. Find (dy)/(dx), when: (cosx)^(y)=(cosy)^(x)

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  4. Find (dy)/(dx), when: (tanx)^(y)=(tany)^(x)

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  5. If y=x^(logx)+(logx)^x then find (dy)/(dx)

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  6. if y=(sin^(- 1)x)/(sqrt(1-x^2)), prove that (1-x^2)(dy)/(dx)=x y+1

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  7. Ify=sqrt(x+y), prove that (dy)/(dx)=(1)/((2y-1)).

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  8. x^ay^b=(x+y)^(a+b) prove that dy/dx=y/x

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  9. If (x^(x)+y^(x))=1," show that "(dy)/(dx)=-{(x^(x)(1+logx)+y^(x)(logy)...

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  10. If y=e^(sinx)+(tanx)^(x)," prove that "(dy)/(dx)=e^(sinx)cosx+(tanx)^(...

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  11. If y=log[x+sqrt((1+x^2)]], prove that sqrt((1+x^2)) (dy)/(dx)=1.

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  12. "If "y=logsinsqrt(x^(2)+1)," prove that "(dy)/(dx)=(x cotsqrt(x^(2)+1)...

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  13. "If "y=logsqrt((1-cosx)/(1+cosx))", show that "(dy)/(dx)="cosec x".

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  14. If y=logtan(pi/4+x/2),\ show that (dy)/(dx)=secxdot Also find the ...

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  15. is y=sqrt((1-sin2x)/(1+sin2x)), show that (dy)/(dx)+sec^2(pi/4-x)=0

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  16. "If "y=logsqrt((1+cos^(2)x)/(1-e^(2x)))", show that "(dy)/(dx)=(e^(2x)...

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  17. "If "y=(x)^(cosx)+(sinx)^(tanx)", prove that "(dy)/(dx)=x^(cosx){(cosx...

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  18. "If "y=(sinx)^(cosx)+(cosx)^(sinx)", prove that "(dy)/(dx)=(sinx)^(cos...

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  19. "If "y=(tanx)^(cotx)+(cotx)^(tanx)",prove that "(dy)/(dx)=(tanx)^(cotx...

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  20. "If "y=x^(cos)+(cosx)^(x)", prove that "(dy)/(dx)=x^(cosx).{(cosx)/(x)...

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