Question

.Find dy/dx by implicit differentiation.y^5 +x^2y^3 = 1 +ye^x2

Answer 1

(2xye^x² - 2xy³) / (5y^4 + 3y²x² - e^x²)

Explanation:

differentiate each term:

d/dx (y^5) + d/dx (x² y³) = d/dx (1) + d/dx (ye^x²)

any time y is differentiated, make sure to include dy/dx:

5y^4 dy/dx + 2xy³ + 3y²x² dy/dx = 0 + e^x² dy/dx + 2xye^x²

collect terms with dy/dx:

dy/dx (5y^4 + 3y²x² - e^x²) = 2xye^x² - 2xy³

divide both sides by (5y^4 + 3y²x² - e^x²):

dy/dx = (2xye^x² - 2xy³) / (5y^4 + 3y²x² - e^x²)