Answer
$f'(x)=xe^{3x}(3x+2)$
$f''(x)=e^{3x}(9x^{2}+12x+2)$
$f'''(x)=9e^{3x}(3x^{2}+6x+2)$
Work Step by Step
$f(x)=x^{2}e^{3x}$
Start the differentiation process by using the product rule:
$f'(x)=(x^{2})(e^{3x})'+(x^{2})'(e^{3x})=...$
Evaluate the derivatives indicated:
$...=3x^{2}e^{3x}+2xe^{3x}=...$
Take out common factor $xe^{3x}$ and simplify:
$...=xe^{3x}(3x+2)$
The expression above is the first derivative. Apply the product rule again to begin the process of finding the second derivative:
$f''(x)=(xe^{3x})(3x+2)'+(3x+2)(xe^{3x})'=...$
Evaluate the derivatives indicated. Use the product rule to evaluate $(xe^{3x})'$:
$...=(xe^{3x})(3)+(3x+2)[x(e^{3x})'+(x)'e^{3x}]=...$
Evaluate the remaining indicated derivatives:
$...=3xe^{3x}+(3x+2)(3xe^{3x}+e^{3x})=...$
Take out common factor $e^{3x}$ from $(3xe^{3x}+e^{3x})$:
$...=3xe^{3x}+e^{3x}(3x+2)(3x+1)=...$
Take out common factor $e^{3x}$ and simplify:
$...=e^{3x}[3x+(3x+2)(3x+1)]=e^{3x}(9x^{2}+12x+2)$
The expression above is the second derivative. Apply the product rule one more time to begin the process of finding the third derivative:
$f'''(x)=e^{3x}(9x^{2}+12x+2)'+(9x^{2}+12x+2)(e^{3x})'=...$
Evaluate the indicated derivatives:
$...=e^{3x}(18x+12)+3e^{3x}(9x^{2}+12x+2)=...$
Take out common factor $e^{3x}$ and simplify:
$...=e^{3x}[3(9x^{2}+12x+2)+18x+12]=...$
$...=e^{3x}(27x^{2}+54x+18)=9e^{3x}(3x^{2}+6x+2)$
The expression above is the third derivative.
