What is my next step after the chain rule

Question

How do I find the implicit derivative of $(5x-y)^4 + 2y^3 = 1294$ I thought I would use the chain rule: $4(5x-y)^3 (5) + 6y^2 = 1294$ so now I got $20(5x-y)^3 + 6y^2 = 1294$ ?? what's next ?

Answer 1

Realize that in implicit differentiation, $y$ is an implicit function of $x$, meaning there's an equation relating $x$ and $y$. What you're doing right now is treating $y$ as a constant, even though it's an implicit function of $x$. Starting with this:

$$(5x-y)^4+2y^3=1294$$

Implicitly differentiate each side:

$$4(5x-y)^3\cdot\,\left(5-{dy\over dx}\right) + 6y^2\cdot {dy\over dx} = 0$$

Notice that instead of treating the derivative of $y$ as $0$, it becomes $\frac{dy}{dx}$. Then solve for ${dy \over dx}$.

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Note that what you're doing is essentially taking the partial derivative (though I use that very loosely), where one variable ($y$) of a function is held constant the function is differentiated with respect to the other, $x$. This is not how you implicitly differentiate; $y$ is an implicit function $x$.