難懂differentiation-答完即送分

1. ylnx = xey

Differentiate both sides with respect to x

yd(lnx)/dx + lnx dy/dx = xd(ey)/dy dy/dx + ey dx/dx

y/x + lnx dy/dx = xey dy/dx + ey

dy/dx = (ey - y/x) / (lnx - xey)

= (xey - y) / (xlnx - x2ey)


2. ln(xy) + x = 4

Differentiate both sides with respect to x

dln(xy)/dx + dx/dx = 0

1/xy (xdy/dx + y) + 1 = 0

xdy/dx + y = -xy

dy/dx = -y(x + 1)/x


3. xey + y = 13

Differentiate both sides with respect to x

xd(ey)/dy dy/dx + ey dx/dx + dy/dx = 0

(xey + 1) dy/dx + ey = 0

dy/dx = -ey / (xey + 1)

dy/dx = -1/(x + e-y)


4. 4x2 + 9y2 = 16

Differentiate both sides with respect to x

4d(x2)/dx + 9d(y2)/dy dy/dx = 0

8x + 18y dy/dx = 0

dy/dx = -4x / 9y