How do you find the derivative of #(x^2)(e^x)#?

Question

Emily Ammerman · Accepted Answer

#e^x(x^2+2x)#

Let #f(x)=x^2# and #g(x)=e^x#. Since we have a product of functions, the derivative can be found with the Product Rule

#f'(x)g(x)+f(x)g'(x)#

From some basic derivatives, we know #f'(x)=2x# and #g'(x)=e^x#. We can now plug these into the Product Rule to get

#2xe^x+x^2e^x#

We can factor out an #e^x# to get

#e^x(x^2+2x)#

I hope this is useful.

Charles Anctil · Answer

undefined To find the derivative of (x^2)(e^x), you can use the product rule of differentiation, which states that if you have two functions u(x) and v(x), then the derivative of their product u(x)v(x) with respect to x is given by the formula: (u(x)v'(x)) + (u'(x)v(x)), where u'(x) and v'(x) are the derivatives of u(x) and v(x) respectively. Applying this rule to the functions x^2 and e^x, you get:

(u(x)) = x^2
(v(x)) = e^x

Now, find the derivatives of u(x) and v(x):

(u'(x)) = 2x (derivative of x^2 using the power rule)
(v'(x)) = e^x (derivative of e^x, which is e^x itself)

Now apply the product rule:

(u(x)v'(x)) + (u'(x)v(x)) = (x^2)(e^x) + (2x)(e^x)

So, the derivative of (x^2)(e^x) with respect to x is (x^2)(e^x) + (2x)(e^x).