How do you find the derivative of #Y= x^2 ( 2x + 3 )#?

1 Answer
Aug 4, 2015

Answer:

We get #y' = 6x^2 +6x#

Explanation:

Method 1 Use the product ruel then simplify.

The product rule tells us that the derivative of a product of two function (I think of them as the First and the Second) is given by:

#d/dx(FS) = F'S+FS'" "#

(Because both addition and multiplication of functions are commutative, other orders are possible.)

So we get (including detail you might prefer to leave out eventually)

#y = x^2(2x+3)#

#y' = [d/dx(x^2)] (2x+3) + x^2[d/dx(2x+3)]" "#

(usually we'll omit writing this step, but we need to DO this)

#y' = [2x] (2x+3)+x^2[2]#

# = 4x^2+6x+2x^2#

# = 6x^2 +6x#

Method 2 Multiply first, the differentiate.

#y = x^2(2x+3)#

#y = 2x^3+3x^2" "# (by algebra)

Now we do not need the product rule, just the sum and power and constant multiple ruel)

#y' = 6x^2 +6x#

Two lessons:

We can use either method to get to the correct answer. (There are many paths to one destination.)

We can take control of how a problem is written. (Unless our tester has told us we must use a particular method -- that is sometimes done to test our knowledge of that method.)