Answer to Question #120664 in Calculus for moreen

Question

Use the techniques of differentiation to find dy over dx
(i) 3e to the power x = xy+xsquared +ysquared
(ii) y= 2xcubed over (xsquared - 4)squared

Answer 1

Hello!

(i) We have,

• 3e^x = xy + x² + y²

Differentiating both sides with respect to x,we get

• 3e^x = (x*dy/dx + y) + 2x +2y*dy/dx ..... (Applying chain rule for xy and y²)
• 3e^x = x*dy/dx + 2y*dy/dx +2x + y
• 3e^x - 2x - y = (dy/dx)*(x+2y)
• dy/dx = (3e^x -2x - y)/(x + 2y) ....(answer)

(ii) We have,

• y = 2x³/(x² - 4)² ..... (i)

The question is in y=u/v form and we will use the formula(dy/dx = (v*du/dx - u*dv/dx)/v²)

So here u = 2x³ and v = (x² - 4)²

Therefore, du/dx = 6x² and dv/dx = 2(x² - 4)*2x = 4x(x² - 4) ....(applying chain rule)

Differentiating equation (i) with respect to x

• dy/dx = (v*du/dx - u*dv/dx)/v²

Putting values of u,v,du/dx,dv/dx, we get

• dy/dx = (x² - 4)²*6x² - 2x³*4x(x² - 4)/(x² - 4)⁴
• dy/dx = (x² - 4)²*6x² - 8x⁴(x² - 4)/(x² - 4)⁴
• dy/dx = x²(x² - 4)*(6(x² - 4) - 8x²)/(x² - 4)⁴
• dy/dx = x²*(6x² - 24 - 8x²)/(x² - 4)³
• dy/dx = x²*( -24 - 2x²)/(x² - 4)³
• dy/dx = -2x²*( x² + 24)/(x² - 4)³

Hence, dy/dx = -2x²*( x² + 24)/(x² - 4)³ ......(answer)