Question

$y³ y² - 5y - x² = -4$ Find the derivative

A) $y' = 3y² + 2y - 5$
B) $y' = 3y² - 2y - 5$
C) $y' = 3y² - 2y + 5$
D) $y' = 3y² + 2y + 5$

Answer 1

Upon simplification and differentiation, the correct derivative of the function $y^5 - 5y - x^2 = -4$ is $y' = \frac{2x}{5y^4 - 5}$. Thus, the correct answer is not provided in the options A, B, C, or D.

Step-by-step explanation:

The question involves finding the derivative of a given function. First, let's simplify the given function:

$y^3 y^2 - 5y - x^2 = -4$

Rewrite the function by combining like terms:

$y^5 - 5y - x^2 = -4$

To find the derivative of the function with respect to $x$, assuming $y$ is a function of $x$, we need to take the derivative term-by-term. We can ignore the constant (-4) as its derivative is 0:

$\frac{d}{dx}(y^5) - \frac{d}{dx}(5y) - \frac{d}{dx}(x^2)$

Using the chain rule for the $y^5$ term and the power rule for the $x^2$ term, we get:

$5y^4y' - 5y' - 2x$

To isolate $y'$ (the derivative of $y$ with respect to $x$), we factor it out:

$y'(5y^4 - 5) = 2x$

Finally, divide both sides by $(5y^4 - 5)$ to solve for $y'$:

$y' = \frac{2x}{5y^4 - 5}$

Therefore, none of the provided answer options (A, B, C, D) are correct.