Find the Derivative - d/dx square root of xy

Problem

d(√(,x*y))/d(x)

Solution

1. Identify the function as a composition requiring the chain rule and implicit differentiation, treating y as a function of x

2. Rewrite the square root as a power to make it easier to differentiate.

√(,x*y)=(x*y)(1/2)

3. Apply the power rule and the chain rule by differentiating the outer function and then multiplying by the derivative of the inner function x*y

d(x*y)/d(x)=1/2*(x*y)(−1/2)⋅(d(x)*y)/d(x)

4. Apply the product rule to differentiate the inner expression x*y

(d(x)*y)/d(x)=xd(y)/d(x)+y(1)

5. Substitute the result of the product rule back into the expression.

1/(2√(,x*y))⋅(xd(y)/d(x)+y)

6. Distribute the term to simplify the final expression.

(xd(y)/d(x)+y)/(2√(,x*y))

Final Answer

d(√(,x*y))/d(x)=(xd(y)/d(x)+y)/(2√(,x*y))

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