Home > Calculus > Differentiating Logarithmic Functions with Base e

What is the derivative of #(lnx)^(3x)#?

Answer 1

We can go by using chain rule and also the rule to derivate exponential functions.

Chain rule states that #(dy)/(dx)=(dy)/(du)(du)/(dx)#. We'll also need the rule to derivate exponential functions, which is: be #f(x)=a^u#, then #f'(x)=a^u(lna)u'#

Let's rename #u=lnx#.

Solving:

#(dy)/(dx)=(lnx)^(3x)(ln(lnx))*3#

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Answer 2

Isabella Ackerman

To find the derivative of ( (\ln(x))^{3x} ), we'll use the chain rule.

Let ( u = \ln(x) ).

Then, ( y = u^{3x} ).

Now, applying the chain rule:

[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ]

[ \frac{dy}{du} = 3xu^{3x-1} ]

[ \frac{du}{dx} = \frac{1}{x} ]

Substitute these into the chain rule:

[ \frac{dy}{dx} = 3x(\ln(x))^{3x-1} \cdot \frac{1}{x} ]

[ \frac{dy}{dx} = 3(\ln(x))^{3x-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.