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Question

If x16 y9=x2+y17, prove that xdydx=2 y

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Solution

We have, x16y9=x2+y17
Taking log on both sides,
logx16y9=logx2+y1716logx+9logy=17logx2+y
Differentiating with respect to x using chain rule,
16ddxlogx+9ddxlogy=17ddxlogx2+y16x+9ydydx=17x2+yddxx2+y16x+9ydydx=17x2+y2x+dydx9ydydx-17x2+ydydx=34xx2+y-16xdydx9y-17x2+y=34xx2+y-16xdydx9x2+y-17yyx2+y=34x2-16x2+yxx2+ydydx9x2+9y-17yyx2+y=34x2-16x2-16yxx2+ydydx9x2-8yyx2+y=18x2-16yxx2+ydydx=yx29x2-8y9x2-8ydydx=2yxxdydx=2y

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