Question

How do you differentiate `sqrt(2x+1)(x^2+1)`?

Answer 1

Break it down into pieces, and derive each using rules you already know...

Explanation:

Start by recognizing that what you have is the product of 2 functions.

The rule for finding the derivative of the product of 2 functions is:

`d/dx(f(x)g(x)) = d/dxf(x) * g(x) + f(x) * d/dx(g(x))` (we'll label this as eq. 1)

here, `f(x) = sqrt(2x + 1)`, and `g(x) = x^2 + 1`

You can use the chain rule for deriving f(x). Start by re-writing it as

`f(x) = (2x + 1)^(1/2)`

`(df)/dx = (1/2)(2x + 1)^(-1/2) * 2 = 1/sqrt(2x + 1)`

The derivative of `g(x) = x^2 + 1` is simple - it's just `2x`

Plug all this back into eq. 1:

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

...which is your answer, but your instructor will probably want you to simplify, because they're like that.

Multiply the right second term by `sqrt(2x + 1)/sqrt(2x + 1)` to get a common denominator:

`= (x^2 + 1)/sqrt(2x + 1) + ((2x + 1) * 2x)/sqrt(2x + 1)`

`= (x^2 + 1 + 4x^2 + 2x)/sqrt(2x + 1)`

`= (5x^2 + 2x + 1)/sqrt(2x + 1)`

...and turn that puppy in.

GOOD LUCK