Find the Derivative - d/d@VAR f(x)=x^(5/2)e^x

Problem

d()/d(x)*x(5/2)*ex

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=x(5/2) and v(x)=ex

2. Apply the product rule, which states d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the individual components using the power rule for x(5/2) and the exponential rule for ex

4. Substitute the derivatives back into the product rule formula.

(d(x(5/2))*ex)/d(x)=x(5/2)d(ex)/d(x)+exd(x(5/2))/d(x)

5. Calculate the derivatives of each part.

d(ex)/d(x)=ex

d(x(5/2))/d(x)=5/2*x(3/2)

6. Combine and simplify the expression by factoring out common terms.

(d(x(5/2))*ex)/d(x)=x(5/2)*ex+5/2*x(3/2)*ex

(d(x(5/2))*ex)/d(x)=x(3/2)*ex*(x+5/2)

Final Answer

(d(x(5/2))*ex)/d(x)=x(3/2)*ex*(x+5/2)

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