Greetings!
In this question, we will be finding the derivative of the given expression.
In this case, we will be using the product rule:
[tex]\frac{d}{dx}[f(x)g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)][/tex]
Whereas:
Solve for the derivative:
[tex]\frac{d}{dx}[x^2*e^x*sin(x)]=x^2\frac{d}{dx}[e^x*sin(x)]+e^x*sin(x)\frac{d}{dx}[x^2][/tex] (1)
The derivates of each component:
[tex]\frac{d}{dx}[x^2]=2x[/tex]
The derivative of x^2 was done using the power rule: [tex]\frac{d}{dx} [x^n]=n*x^{(n-1)[/tex]
[tex]\frac{d}{dx}[e^x]=e^x\\\\\frac{d}{dx}[sin(x)]=cos(x)[/tex]
Using the product rule again for [tex]\frac{d}{dx}[e^x*sin(x)][/tex]
[tex]\frac{d}{dx}[e^x*sin(x)]\\\\f(x)=e^x\\\\g(x)=sin(x)\\\\\frac{d}{dx}[e^x*sin(x)]=e^x\frac{d}{dx}[sin(x)]+sin(x)\frac{d}{dx}[e^x]=e^xcos(x)+sin(x)e^x[/tex]
Now, finish solving for the derivative by plugging in our found values into equation (1)
[tex]\frac{d}{dx}[x^2*e^x*sin(x)]=x^2\frac{d}{dx}[e^x*sin(x)]+e^x*sin(x)\frac{d}{dx}[x^2]\\\\=x^2(e^xcos(x)+sin(x)e^x)+e^xsin(x)*2x[/tex]
Answer:
[tex]\boxed{\frac{d}{dx}[x^2*e^x*sin(x)]=x^2(e^xcos(x)+sin(x)e^x)+2x*e^xsin(x)}[/tex]
