How do you find the second derivative of #z=xsqrt(x+1)#?
Calculate the first derivative using the product rule:
Simplifying:
Differentiate again using the quotient rule:
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To find the second derivative of ( z = x\sqrt{x+1} ), follow these steps:

Start by finding the first derivative of ( z ) with respect to ( x ) using the product rule.
[ \frac{dz}{dx} = \sqrt{x+1} + x\left(\frac{1}{2\sqrt{x+1}}\right) ]

Simplify the expression:
[ \frac{dz}{dx} = \sqrt{x+1} + \frac{x}{2\sqrt{x+1}} ]

Then differentiate ( \frac{dz}{dx} ) with respect to ( x ) to find the second derivative.
[ \frac{d^2z}{dx^2} = \frac{1}{2\sqrt{x+1}} + \frac{1}{2\sqrt{x+1}}  \frac{x}{4(x+1)^{\frac{3}{2}}} ]

Combine like terms and simplify further:
[ \frac{d^2z}{dx^2} = \frac{1}{\sqrt{x+1}}  \frac{x}{2(x+1)^{\frac{3}{2}}} ]
So, the second derivative of ( z ) with respect to ( x ) is ( \frac{1}{\sqrt{x+1}}  \frac{x}{2(x+1)^{\frac{3}{2}}} ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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