Let `y = (x+1)(x^2+1)`. Find the derivativeby first using the product rule and then by multyplying the factors first and then taking the derivative.

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txmedteach | High School Teacher | (Level 3) Associate Educator

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So, we're going to calculate the derivative in 2 ways.

Calculating the Derivative Based on Product Rule

First, we'll use the product rule, stated here (also, see link below):

Given two functions of x, f(x) and g(x), the following holds true:

`d/(dx)(f*g) = g(df)/(dx)+f(dg)/(dx)`

Put another way,

`(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)`

In this problem, we can define f(x) and g(x) in the following way to find the derivative using the product rule:

`f(x) = x+1`

`g(x) = x^2+1`

Given this information, we can calculate these functions' derivatives:

`(d(f(x)))/(dx)=1`

`(d(g(x)))/dx = 2x`

Now, we can use the product rule to find the derivative of y(x) above:

`dy/dx = d/dx(f(x)g(x)) = (1)(x^2+1) + (x+1)(2x)`

Simplifying:

`dy/dx = x^2 + 1 + 2x^2 + 2x`

`dy/dx = 3x^2 + 2x + 1`

And there is the derivative that we calculated based on the product rule.

Calculating Derivative by simplifying the Original Function First

Now, we'll first simplify y(x) before attempting to calculate the derivative.

`y(x) = (x+1)(x^2+1)`

We can use the FOIL method to simplify this function:

`y(x) = x^3 + x + x^2 + 1`

For conveience, let's reorder the terms:

`y(x) = x^3 + x^2 + x + 1`

Now, we can take the derivative fairly easily:

`dy/dx = 3x^2 + 2x + 1`

Thankfully, this is equivalent to the derivative we calculated using the product rule! We know now that we have the right function that represents `dy/dx`.

Sources:

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