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borishaifa [10]
2 years ago
10

Suppose x and y are related by the given equation and use implicit differentiation to determine dydx.x7y+y7x=7.

Mathematics
1 answer:
Liono4ka [1.6K]2 years ago
6 0

Looks like the equation is

x^7y+y^7x=7

Differentiate both sides with respect to x, taking y to be a function of x.

\dfrac{\mathrm d[x^7y+y^7x]}{\mathrm dx}=\dfrac{\mathrm d[7]}{\mathrm dx}

\dfrac{\mathrm d[x^7]}{\mathrm dx}y+x^7\dfrac{\mathrm dy}{\mathrm dx}+\dfrac{\mathrm d[y^7]}{\mathrm dx}x+y^7\dfrac{\mathrm dx}{\mathrm dx}=0

7x^6y+x^7\dfrac{\mathrm dy}{\mathrm dx}+7y^6x\dfrac{\mathrm dy}{\mathrm dx}+y^7=0

Solve for \frac{\mathrm dy}{\mathrm dx}:

(x^7+7y^6)\dfrac{\mathrm dy}{\mathrm dx}=-(7x^6y+y^7)

\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac{7x^6y+y^7}{x^7+7y^6}

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