Implicit differentiation for the given equation

Question

Determine the first and second derivative of $y$ being given the equation $x^5+y^5-15=0$

Answer 1

Consider the implicit function $$F=x^5+y^5-15=0$$ Differentiate $$F'_x=5x^4\qquad F'_y=5y^4\implies \frac{dy}{dx}=-\frac{F'_x}{F'_y}=-x^4y^{-4}$$ $$\frac{d^2y}{dx^2}=-4x^3y^{-4}+4x^{4}y^{-5}\frac{dy}{dx}$$ and continue.

Answer 2

The basic idea is: (1) Take derivatives of the equation (2) solve for $y'$ resp. $y''$. If you have to take the derivative of some function $g(y)$ with respect to $x$, recall that by the chain rule, we have $$ \bigl(g(y)\bigr)' = g'(y)y' $$ Hence, $$ (y^5)' = 5y^4y' $$ Taking the derivative of your equation gives $$ 5y^4y' + 5x^4 = 0 \tag+$$ Now solve for $y'$. Taking another derivative of $(+)$ gives us $$ 20y^3(y')^2 + 5y^4y'' + 20x^3 = 0 $$ Pluging in $y'$ from above and solving for $y''$ gives us the second derivative. I'm sure, you can do the rest.