How do you differentiate #f(x)=(x^(1/2))(lnx)#?

Answer 1

#1/(2sqrtx) lnx+ 1/sqrtx#

This needs to be differentiated using the product rule:

#d/dxf(x)g(x)=g(x)f'(x)+f(x)g'(x)#
#d/dx x^(1/2)= 1/2x^(-1/2)= 1/(2sqrtx)#
#d/dxlnx= 1/x#
#d/dxf(x)=1/(2sqrtx)lnx+sqrtx/x=1/(2sqrtx)+1/sqrtx#
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Answer 2

To differentiate ( f(x) = \sqrt{x} \cdot \ln(x) ), you can use the product rule.

The product rule states that if you have two functions, ( u(x) ) and ( v(x) ), then the derivative of their product, ( u(x) \cdot v(x) ), is given by:

[ (u \cdot v)' = u' \cdot v + u \cdot v' ]

So for ( f(x) = \sqrt{x} \cdot \ln(x) ):

Let ( u(x) = \sqrt{x} ) and ( v(x) = \ln(x) ).

Then, ( u'(x) = \frac{1}{2\sqrt{x}} ) (using the power rule for differentiation) and ( v'(x) = \frac{1}{x} ) (using the derivative of ln(x)).

Now apply the product rule:

[ f'(x) = \frac{1}{2\sqrt{x}} \cdot \ln(x) + \sqrt{x} \cdot \frac{1}{x} ]

Simplify this expression to get the derivative of ( f(x) ).

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Answer 3

To differentiate ( f(x) = \sqrt{x} \cdot \ln(x) ), we can use the product rule, which states that if ( f(x) = u(x) \cdot v(x) ), then ( f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) ).

Let ( u(x) = \sqrt{x} ) and ( v(x) = \ln(x) ). Then, we have:

( u'(x) = \frac{1}{2\sqrt{x}} ) (using the power rule)

( v'(x) = \frac{1}{x} ) (using the derivative of ln(x))

Now, applying the product rule:

( f'(x) = \frac{1}{2\sqrt{x}} \cdot \ln(x) + \sqrt{x} \cdot \frac{1}{x} )

( f'(x) = \frac{\ln(x)}{2\sqrt{x}} + \frac{\sqrt{x}}{x} )

This can be simplified by combining the terms:

( f'(x) = \frac{\ln(x) + 2}{2\sqrt{x}} )

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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