Question

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

Answer 1

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 $\frac{\mathrm dy}{\mathrm dx}$:

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

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