What is the derivative of y = 2x^4 * e^x in its simplest form?

To determine the derivative of the function ( y = 2x^4 e^x ), we will apply the product rule for differentiation. The product rule states that if you have two functions multiplied together, say ( u ) and ( v ), the derivative is given by ( \frac{d(uv)}{dx} = u'v + uv' ).

In this case, let:

• ( u = 2x^4 ) and

• ( v = e^x ).

Now, we need to find the derivatives of both functions:

• The derivative of ( u ) (which is ( 2x^4 )) is ( u' = 8x^3 ).

• The derivative of ( v ) (which is ( e^x )) is ( v' = e^x ).

Next, applying the product rule:

[

y' = u'v + uv' = (8x^3)(e^x) + (2x^4)(e^x).

]

We can factor out the common term ( e^x ):

[

y' = e^x (8x^3 + 2x^4).