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

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Answer 1 · 2017-08-31T20:20:07

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

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

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