I've been asked to differentiate the equation $x^7 + 7xy^2 = 7$ with respect to the variable $t$ and express $\frac{dy}{dt}$ in terms of $\frac{dx}{dt}$. I tried to solve it like this:
$\frac{d}{dt}(x^7) + 7\frac{d}{dt}(xy^2) = \frac{d}{dt}(7)$
Using the product rule: $7x^6\frac{dx}{dt} + 7(\frac{d}{dt}(x)y^2 + x\frac{d}{dt}(y^2)) = 0$
$7x^6\frac{dx}{dt} + 7(y^2\frac{dx}{dt} + 2xy\frac{dy}{dt}) = 0$
Factoring out the 7: $x^6\frac{dx}{dt} + y^2\frac{dx}{dt} + 2xy\frac{dy}{dt} = 0$
Then isolating $\frac{dy}{dt}$ on one side: $\frac{dy}{dt} = -\frac{x^6\frac{dx}{dt} - y^2\frac{dx}{dt}}{2xy}$
WebWork tells me this is incorrect and wolframalpha has been no help since it doesn't understand how to differentiate with respect to a variable not in the function (or I don't know how to input it into wolframalpha).
Am I misunderstanding the question or is there a mistake in my algebra?
$$\dfrac{dy}{dt} = -\dfrac{\dfrac{dx}{dt}(x^6 + y^2)}{2xy}$$