How do you differentiate #f(x)=(x^(1/2))(lnx)#?
This needs to be differentiated using the product rule:
By signing up, you agree to our Terms of Service and Privacy Policy
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) ).
By signing up, you agree to our Terms of Service and Privacy Policy
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}} )
By signing up, you agree to our Terms of Service and Privacy Policy
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.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7