Explanation
1. Simplify the Given Equation
The given equation is (ey)x−y=0. By the power rule for exponents (ab)c=abc, we can simplify the equation as follows:
exy−y=0
2. Differentiate Implicitly with Respect to x
To find dxdy, we differentiate both sides of the equation with respect to x using the chain rule and product rule:
dxd(exy)−dxd(y)=0
Using the chain rule for the first term:
exy⋅dxd(xy)−dxdy=0
Apply the product rule to dxd(xy):
exy⋅(y+xdxdy)−dxdy=0
3. Solve for dxdy
From the original equation exy−y=0, we know that exy=y. Substituting this into our derivative expression:
y⋅(y+xdxdy)−dxdy=0
Distribute y:
y2+xydxdy−dxdy=0
Rearrange the terms to isolate dxdy:
y2=dxdy−xydxdy
y2=dxdy(1−xy)
dxdy=1−xyy2
Conclusion: The correct expression for dxdy is 1−xyy2.
Correct Option: (c)
