Find the Derivative of:
F(x)= (x^2) (e^x) + (e^x)
would I use the product rule?
F'(x)= 2x(e^x) + (e^x) (x^2) + (e^x)
= (e^x) ( x^2 + 2x + 1 )
yes, you used the product rule correctly for the first term.
I wouldn't be surprised if your text factored the answer to get
e^x(x+1)^2
Oh okay thank you!
Yes, to differentiate the function F(x) = (x^2)(e^x) + (e^x), you would indeed use the product rule. The product rule states that if you have a function of the form f(x) = u(x)v(x), where u(x) and v(x) are both differentiable functions of x, then the derivative of f(x) is given by:
f'(x) = u'(x)v(x) + u(x)v'(x)
In this case, let's assign u(x) = x^2 and v(x) = e^x. Now, we need to find the derivatives of u(x) and v(x):
u'(x) = 2x
v'(x) = e^x
Now, we can apply the product rule to find the derivative of F(x):
F'(x) = u'(x)v(x) + u(x)v'(x)
= 2x(e^x) + (x^2)(e^x) + e^x
= (e^x)(x^2 + 2x + 1)
Therefore, the derivative of F(x) is (e^x)(x^2 + 2x + 1).