If xy (x + y) = 2 , then find dy/dx
Answers
Answer:
dy / dx = - ( 2xy + y² ) / ( x² + 2xy )
Step-by-step explanation:
Given---> xy ( x + y ) = 2
To find ---> Derivative of given expression
Solution---> We have some formulee of differentiation , as follows,
1) d/dx ( u v ) = u dv/dx + v du/dx
2) d/dx ( x ) = 1
3) d/dx ( y ) = dy / dx
4) d/dx ( x² ) = 2x
5) d/dx ( Constant ) = 0
Now , ATQ,
xy ( x + y ) = 2
=> x²y + xy² = 2
Differentiating with respect to x , we get,
=> d/dx ( x²y ) + d/dx ( xy² ) = d/dx ( 2 )
=> x² dy/dx + y d/dx ( x² ) + x d/dx( y² ) + y² d/dx (x)
= 0
=> x² dy/dx + y (2x ) + x (2y) dy/dx + y² ( 1 ) = 0
=> x² dy/dx + 2xy + 2xy dy/dx + y² = 0
=> ( x² + 2xy ) dy/dx = - 2xy - y²
=> ( x² + 2xy ) dy/dx = - ( 2xy + y² )
=> dy / dx = - ( 2xy + y² ) / ( x² + 2xy )
=> dy / dx = - y ( 2x + y ) / x ( x + 2y )
Additional information--->
1) d/dx ( xⁿ ) = nxⁿ⁻¹
2) d/dx ( eˣ ) = eˣ
3) d/dx ( aˣ ) = aˣ loga
4) d/dx ( logx ) = 1/x
5) d/dx ( Sinx ) = Cosx
6) d/dx ( Cosx ) = -Sinx
7) d/dx ( tanx ) = Sec²x
8) d/dx ( Secx ) = Secx tanx
9) d/dx ( Cotx ) = - Cosec²x
10) d/dx ( Cosecx ) = - Cosecx Cotx
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Answer:
To find ---> Derivative of given expression
Solution---> We have some formulee of differentiation , as follows,
1) d/dx ( u v ) = u dv/dx + v du/dx
2) d/dx ( x ) = 1
3) d/dx ( y ) = dy / dx
4) d/dx ( x² ) = 2x
5) d/dx ( Constant ) = 0
Now , ATQ,
xy ( x + y ) = 2
=> x²y + xy² = 2
Differentiating with respect to x , we get,
=> d/dx ( x²y ) + d/dx ( xy² ) = d/dx ( 2 )
=> x² dy/dx + y d/dx ( x² ) + x d/dx( y² ) + y² d/dx (x)
= 0
=> x² dy/dx + y (2x ) + x (2y) dy/dx + y² ( 1 ) = 0
=> x² dy/dx + 2xy + 2xy dy/dx + y² = 0
=> ( x² + 2xy ) dy/dx = - 2xy - y²
=> ( x² + 2xy ) dy/dx = - ( 2xy + y² )
=> dy / dx = - ( 2xy + y² ) / ( x² + 2xy )
=> dy / dx = - y ( 2x + y ) / x ( x + 2y )
Additional information--->
1) d/dx ( xⁿ ) = nxⁿ⁻¹
2) d/dx ( eˣ ) = eˣ
3) d/dx ( aˣ ) = aˣ loga
4) d/dx ( logx ) = 1/x
5) d/dx ( Sinx ) = Cosx
6) d/dx ( Cosx ) = -Sinx
7) d/dx ( tanx ) = Sec²x
8) d/dx ( Secx ) = Secx tanx
9) d/dx ( Cotx ) = - Cosec²x
10) d/dx ( Cosecx ) = - Cosecx Cotx
Step-by-step explanation: