# Higher Derivatives

• Mar 4th 2009, 12:01 PM
Dickson
Higher Derivatives
1) Find the first and second derivatives:

y = Square Root((x^2) + 1)

2) If f(x) = Square Root(1 + (x^3)) , find the second derivative (2). So x = 2.
• Mar 4th 2009, 12:14 PM
HallsofIvy
$y= (x^2+ 1)^{1/2}$

Apply the power rule, $(u^n)'= nu^{n-1}$ and the chain rule.
• Mar 4th 2009, 03:56 PM
Dickson
Did that and got the wrong answer
• Mar 4th 2009, 06:09 PM
HallsofIvy
Then show what you did, please.
• Mar 4th 2009, 07:40 PM
Dickson
First Deriv. = (1/2)(x^2 + 1) ^ (-1/2) (2x)
= x(x^2 + 1) ^ (-1/2)

Second = (-1/2x) (x^2 + 1) ^ (-3/2) (2x)

= -x^2 (x^2 + 1) ^ (-3/2)

Book says it should be only an x in the front
• Mar 5th 2009, 03:25 AM
mr fantastic
Quote:

Originally Posted by Dickson
First Deriv. = (1/2)(x^2 + 1) ^ (-1/2) (2x)
= x(x^2 + 1) ^ (-1/2)

Second = (-1/2x) (x^2 + 1) ^ (-3/2) (2x)

= -x^2 (x^2 + 1) ^ (-3/2)

Book says it should be only an x in the front

To get the second derivative you have to use either the product or quotient rule as well as the chain rule.

I suggest writing the first derivative as $\frac{x}{(x^2 + 1)^{1/2}}$ and using the quotient rule to differentiate. To get the derivative of the denominator the chain rule is required.

A bit of algebra will be required to get the answer in the required form.