Find the Derivative - d/dx y=(x+3)^2e^(4x)

Problem

d()/d(x)*(x+3)2*e(4*x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, u=(x+3)2 and v=e(4*x) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first part u=(x+3)2 using the power rule and chain rule.

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

3. Differentiate the second part v=e(4*x) using the chain rule.

d(e(4*x))/d(x)=4*e(4*x)

4. Apply the product rule formula by substituting the parts.

d(y)/d(x)=(x+3)2*(4*e(4*x))+e(4*x)*(2*(x+3))

5. Factor out the common terms 2 (x+3) and e(4*x) to simplify the expression.

d(y)/d(x)=2*(x+3)*e(4*x)*(2*(x+3)+1)

6. Simplify the expression inside the parentheses.

d(y)/d(x)=2*(x+3)*e(4*x)*(2*x+6+1)

d(y)/d(x)=2*(x+3)*(2*x+7)*e(4*x)

Final Answer

d()/d(x)*(x+3)2*e(4*x)=2*(x+3)*(2*x+7)*e(4*x)

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