Question

Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy. x5y2 − x3y + 5xy3 = 0

Answer 1

Answer:
$\frac{dx}{dy}=\frac{x^3y-2x^5y-15xy^2}{5x^4y^2-3x^2y +5y^3}$

Step-by-step explanation:

The given function is:

$x^5y^2-x^3y+5xy^3=0$.

Use the product rule and differentiate implicitly with respect to y.

$2yx^5+5x^4y^2\frac{dx}{dy}-x^3-3x^2y\frac{dx}{dy}+15xy^2+5y^3\frac{dx}{dy}=0$

$\Rightarrow 5x^4y^2\frac{dx}{dy}-3x^2y\frac{dx}{dy} +5y^3\frac{dx}{dy}=x^3-2x^5y-15xy^2$

$\Rightarrow (5x^4y^2-3x^2y +5y^3)\frac{dx}{dy}=x^3-2x^5y-15xy^2$

$\Rightarrow \frac{dx}{dy}=\frac{x^3-2x^5y-15xy^2}{5x^4y^2-3x^2y +5y^3}$