How do you find the derivative of #3x^(lnx)#?

Answer 1
Let #y=3x^{ln(x)}#, then #ln(y)=ln(3x^{ln(x)})=ln(3)+ln(x^{ln(x)})=ln(3)+(ln(x))^{2}# (using the properties that #ln(AB)=ln(A)+ln(B)# for #A,B > 0# and #ln(A^{p})=pln(A)# for #A > 0#).
Now differentiate #\ln(y)=ln(3)+(ln(x))^{2}# with respect to #x#, keeping in mind that #y# is a function of #x# and using the Chain Rule to get #\frac{1}{y}\cdot \frac{dy}{dx}=2(ln(x))^{1}\cdot \frac{1}{x}=\frac{2\ln(x)}{x}#.
Multiplying both sides of this last equation by #y=3x^{ln(x)}# helps us see that #\frac{dy}{dx}=3x^{ln(x)}\cdot \frac{2\ln(x)}{x}=6x^{ln(x)-1}ln(x)#. This is valid as long as #x > 0#.
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Answer 2

To find the derivative of (3x^{\ln x}), you can use the product rule along with the chain rule. The derivative is:

[ \frac{d}{dx} (3x^{\ln x}) = 3 \frac{d}{dx} (x^{\ln x}) + x^{\ln x} \frac{d}{dx}(3) ]

Apply the chain rule to (x^{\ln x}):

[ \frac{d}{dx} (x^{\ln x}) = x^{\ln x} \left(\frac{d}{dx} (\ln x) + \ln x \frac{d}{dx} (\ln x)\right) ]

[ = x^{\ln x} \left(\frac{1}{x} + \ln x \frac{1}{x}\right) ]

[ = x^{\ln x} \left(\frac{1}{x} + \frac{\ln x}{x}\right) ]

Substitute this back into the derivative equation:

[ 3 \left(x^{\ln x} \left(\frac{1}{x} + \frac{\ln x}{x}\right)\right) + x^{\ln x} \frac{d}{dx}(3) ]

[ = 3x^{\ln x - 1} + 3(\ln x)x^{\ln x - 1} + x^{\ln x} \cdot 0 ]

[ = 3x^{\ln x - 1} + 3(\ln x)x^{\ln x - 1} ]

So, the derivative of (3x^{\ln x}) with respect to (x) is (3x^{\ln x - 1} + 3(\ln x)x^{\ln x - 1}).

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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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