Find the Derivative - d/dx square root of xe^x

Problem

d(√(,x*ex))/d(x)

Solution

1. Rewrite the square root as a power to prepare for the chain rule.

√(,x*ex)=(x*ex)(1/2)

2. Apply the chain rule by differentiating the outer function u(1/2) and then multiplying by the derivative of the inner function x*ex

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

3. Apply the product rule to find the derivative of the inner function x*ex

(d(x)*ex)/d(x)=(1)*(ex)+(x)*(ex)

4. Factor the result of the product rule.

(d(x)*ex)/d(x)=ex*(1+x)

5. Substitute the inner derivative back into the chain rule expression.

d(√(,x*ex))/d(x)=1/2*(x*ex)(−1/2)⋅ex*(1+x)

6. Simplify the expression by moving the negative exponent to the denominator and combining terms.

d(√(,x*ex))/d(x)=(ex*(1+x))/(2√(,x*ex))

Final Answer

d(√(,x*ex))/d(x)=(ex*(1+x))/(2√(,x*ex))

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