Find dxdy, if yx+xy+xx=ab.
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Step-by-Step Solution
Step 1: Decompose the equation
The given equation involves a sum of three complex terms, each of which is a function raised to another function. To simplify the differentiation process, we can break down the equation into individual terms. Let u=yx, v=xy, and w=xx. The equation then becomes u+v+w=ab.
Step 2: Differentiate each term with respect to x
Since ab is a constant (as a and b are constants), its derivative with respect to x is 0. Therefore, we need to find the derivative of each term u, v, and w with respect to x and sum them up to 0.
Step 3: Calculate dxdu
To differentiate u=yx, we take the natural logarithm of both sides: lnu=xlny. Then, we differentiate implicitly with respect to x. Using the product rule on the right side, we get u1dxdu=lny+xy1dxdy. Finally, we multiply by u to solve for dxdu.
Step 4: Calculate dxdv
Similarly, for v=xy, we take the natural logarithm: lnv=ylnx. Differentiating implicitly with respect to x gives v1dxdv=dxdylnx+yx1. Multiplying by v yields the expression for dxdv.
Step 5: Calculate dxdw
For w=xx, we take the natural logarithm: lnw=xlnx. Differentiating implicitly with respect to x gives w1dxdw=lnx+xx1. Multiplying by w yields the expression for dxdw.
Step 6: Substitute and solve for dxdy
Now, substitute the expressions for dxdu, dxdv, and dxdw back into the equation dxdu+dxdv+dxdw=0. Group the terms containing dxdy on one side and the remaining terms on the other side. Finally, divide to solve for dxdy.