How do you differentiate #y = (x^3 + 4)(-x^6 + x^4)#?

The easiest way is to simply multiply the two brackets

And then take the derivative of polinomials like this

You could use the product rule but this is a case where the simplest way is to foil the multiplication and then derive the result.

To differentiate ( y = (x^3 + 4)(-x^6 + x^4) ), you can use the product rule. The product rule states that if you have two functions, ( u(x) ) and ( v(x) ), then the derivative of their product is given by:

[ \frac{d}{dx}(u \cdot v) = u'v + uv' ]

Where ( u' ) and ( v' ) are the derivatives of ( u(x) ) and ( v(x) ) with respect to ( x ), respectively.

So, applying the product rule to ( y = (x^3 + 4)(-x^6 + x^4) ), we get:

[ y' = (3x^2)(-x^6 + x^4) + (x^3 + 4)(-6x^5 + 4x^3) ]

[ y' = -3x^8 + 3x^6 - 6x^8 + 4x^6 - 24x^5 + 16x^3 ]

[ y' = -9x^8 + 7x^6 - 24x^5 + 16x^3 ]

So, the derivative of ( y = (x^3 + 4)(-x^6 + x^4) ) is ( y' = -9x^8 + 7x^6 - 24x^5 + 16x^3 ).