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Let x^4+y^4=16 and consider y as a function of x. use the implicit differentiation to find y"

User Asad Ullah
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4 votes

Answer:


(dy)/(dx) = - \frac{ {x}^(3) }{ {y}^(3) }

Explanation:

Differentiate both sides of the equation (consider y as a function of x).


(d)/(dx) ( {x}^(4) + {y}^(4) (x)) = (d)/(dx) (16)

the derivative of a sum/difference is the sum/difference of derivatives.


( (d)/(dx) ( {x}^(4) + {y}^(4) (x))


= ( (d)/(dx) ( {x}^(4) ) + (d)/(dx) ( {y}^(4) (x)))

the function of y^4(x) is the composition of f(g(x)) of the two functions.

the chain rule:


(d)/(dx) (f(g(x))) = (d)/(du) (f(u)) (d)/(dx) (g(x))


= ( (d)/(du) ( {u}^(4) ) (d)/(dx) (y(x))) + (d)/(dx) ( {x}^(4) )

apply the power rule:


(4 {u}^(3) ) (d)/(dx) (y(x)) + (d)/(dx) ( {x}^(4) )

return to the old variable:


4(y(x) {)}^(3) (d)/(dx) (y(x)) + (d)/(dx) ( {x}^(4) )

apply the power rule once again:


4 {y}^(3) (x) (d)/(dx) (y(x)) + (4 {x}^(3) )

simplify:


4 {x}^(3) + 4 {y}^(3) (x) (d)/(dx) (y(x))


= 4( {x}^(3) + {y}^(3) (x) (d)/(dx) (y(x)))


= (d)/(dx) ( {x}^(4) + {y}^(4) ))


= 4( {x}^(3) + {y}^(3) (x) (d)/(dx)(y(x)))

differentiate the equation:


( (d)/(dx) (16)) = (0)


= (d)/(dx) (16) = 0

derivative:


4 {x}^(3) + 4 {y}^(3) (dy)/(dx) = 0


(dy)/(dx) = - \frac{ {x}^(3) }{ {y}^(3) }

User Shubendrak
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