[tex]f(x)=x^3e^{2x}\\\\f'(x)=(x^3e^{2x})'\qquad\text{use}\ (f\cdot g)'(x)=f'(x)g(x)+f(x)g'(x)\\\\=(x^3)'(e^{2x})+(x^3)(e^{2x})'\\\\\text{use}\ (x^n)'=nx^{n-1}\ \text{and}\ (e^x)'=e^x\ \text{and}\ f'[g(x)]=f'(g(x))\cdot g'(x)\\\\=3x^2e^{2x}+x^3e^{2x}\cdot2\\\\=\boxed{3x^2e^{2x}+2x^3e^{2x}}[/tex]
