Find the Derivative - d/d@VAR f(x)=e^(-x) square root of x

Problem

d()/d(x)*e(−x)√(,x)

Solution

1. Identify the function as a product of two terms, u=e(−x) and v=√(,x) which requires the product rule: d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first term u=e(−x) using the chain rule to get d(e(−x))/d(x)=−e(−x)

3. Differentiate the second term v=√(,x)=x(1/2) using the power rule to get d(x(1/2))/d(x)=1/2*x(−1/2) which is 1/(2√(,x))

4. Apply the product rule formula by substituting the derivatives found in the previous steps.

ƒ(x)′=e(−x)1/(2√(,x))+√(,x)*(−e(−x))

5. Factor out the common term e(−x) to simplify the expression.

ƒ(x)′=e(−x)*(1/(2√(,x))−√(,x))

6. Find a common denominator inside the parentheses to combine the terms.

ƒ(x)′=e((1−2*x)/(2√(,x)))(−x)

Final Answer

d()/d(x)*e(−x)√(,x)=(e(−x)*(1−2*x))/(2√(,x))

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