if -5x^3 + y^2 - x = 0 then find dy/dx in terms of x and y.

Explanation:

Step1: Differentiate each term

Differentiate $-5x^{3}$, $y^{2}$, and $-x$ with respect to $x$. The derivative of $-5x^{3}$ is $-15x^{2}$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$. For $y^{2}$, by the chain - rule, we have $2y\frac{dy}{dx}$. The derivative of $-x$ is $-1$. So, $\frac{d}{dx}(-5x^{3}+y^{2}-x)=\frac{d}{dx}(0)$. $$-15x^{2}+2y\frac{dy}{dx}-1 = 0$$

Step2: Isolate $\frac{dy}{dx}$

First, move the non - $\frac{dy}{dx}$ terms to the other side of the equation: $$2y\frac{dy}{dx}=15x^{2}+1$$ Then, divide both sides by $2y$ (assuming $y\neq0$) to solve for $\frac{dy}{dx}$: $$\frac{dy}{dx}=\frac{15x^{2}+1}{2y}$$

Answer:

$\frac{15x^{2}+1}{2y}$