What is the derivative of #x^(lnx)#?
The derivative of
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To find the derivative of ( x^{\ln(x)} ), you can use logarithmic differentiation.
Let ( y = x^{\ln(x)} ). Take the natural logarithm of both sides:
( \ln(y) = \ln(x^{\ln(x)}) )
Using the properties of logarithms, we get:
( \ln(y) = \ln(x) \cdot \ln(x) )
Now differentiate both sides with respect to ( x ):
( \frac{1}{y} \cdot \frac{dy}{dx} = \frac{1}{x} \cdot \ln(x) + \frac{\ln(x)}{x} )
Substitute ( y = x^{\ln(x)} ) back in:
( \frac{1}{x^{\ln(x)}} \cdot \frac{dy}{dx} = \frac{1}{x} \cdot \ln(x) + \frac{\ln(x)}{x} )
Multiply both sides by ( x^{\ln(x)} ):
( \frac{dy}{dx} = x^{\ln(x)} \left( \frac{1}{x} \cdot \ln(x) + \frac{\ln(x)}{x} \right) )
Simplify:
( \frac{dy}{dx} = x^{\ln(x)} \left( \frac{\ln(x)}{x} + \frac{\ln(x)}{x} \right) )
( \frac{dy}{dx} = x^{\ln(x)} \cdot \frac{2\ln(x)}{x} )
So, the derivative of ( x^{\ln(x)} ) with respect to ( x ) is ( x^{\ln(x)} \cdot \frac{2\ln(x)}{x} ).
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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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