Find the Derivative - d/d@VAR g(x)=5e^x square root of x

Problem

d()/d(x)*5*ex√(,x)

Solution

1. Identify the function as a product of two terms, 5*ex and √(,x) which requires the product rule (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Rewrite the square root term using a fractional exponent to make differentiation easier: √(,x)=x(1/2)

3. Apply the product rule by setting u=5*ex and v=x(1/2)

4. Differentiate each part: (d(5)*ex)/d(x)=5*ex and d(x(1/2))/d(x)=1/2*x(−1/2)

5. Combine the results using the product rule formula: 5*ex*(1/2*x(−1/2))+x(1/2)*(5*ex)

6. Simplify the expression by factoring out common terms like 5*ex and converting negative exponents back to radical form.

7. Finalize the expression by finding a common denominator if necessary.

Final Answer

(d(5)*ex√(,x))/d(x)=5*ex√(,x)+(5*ex)/(2√(,x))

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