Find the Derivative - d/dy y^(3/2)(y+ce^y)

Problem

d()/d(y)*y(3/2)*(y+c*ey)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, u=y(3/2) and v=y+c*ey apply the product rule.

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

3. Differentiate the first part u=y(3/2) using the power rule.

d(y(3/2))/d(y)=3/2*y(1/2)

4. Differentiate the second part v=y+c*ey using the sum rule and the rule for exponential functions.

d(y+c*ey)/d(y)=1+c*ey

5. Substitute these derivatives back into the product rule formula.

d()/d(y)*y(3/2)*(y+c*ey)=y(3/2)*(1+c*ey)+(y+c*ey)*(3/2*y(1/2))

6. Distribute the terms to simplify the expression.

y(3/2)+c*y(3/2)*ey+3/2*y(3/2)+3/2*c*y(1/2)*ey

7. Combine like terms, specifically the y(3/2) terms.

5/2*y(3/2)+c*y(3/2)*ey+3/2*c*y(1/2)*ey

8. Factor out common terms if desired, such as 1/2*y(1/2)

1/2*y(1/2)*(5*y+2*c*y*ey+3*c*ey)

Final Answer

d()/d(y)*y(3/2)*(y+c*ey)=5/2*y(3/2)+c*y(3/2)*ey+3/2*c*y(1/2)*ey

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