Find the derivatives of `y=(x^2+1)(x^3+3)`.

Explanation

Using the product Rule with `u = x^2 + 1 and v = x^3 + 3`, we find
`d/(dx)[(x^2 + 1)(x^2 + 3)] = (x^2 + 1)(3x^2)+(x^2 + 3)(2x) = 3x^4 + 3x^2 + 2x^4 + 6x = 5x^4 + 3x^2 + 6x`
Example can be done as well (perhaps better) by multiplying out the original expression for y and differentiating the resulting polynomial. We now check :
`y = (x^2 + 1) (x^3 + 3) = x^5 + x^3 + 3x^2 + 3`
`(dy)/(dx) = 5x^4 + 3x^2 + 6x`
This is in agreement with our first calculation. There are times, however, when the product Rule must be used. In the following examples. We have only numerical values to work with.