Find the derivative using logarithmic differentiation?

Answer 1

#d/dx x^(4x) = 4x^(4x)(1+lnx)#

Pose:

#f(x) = x^(4x)#
#g(x) = ln f(x) = 4xlnx#

Then:

#g'(x) = 4+4lnx = (f'(x))/f(x) #

and:

#f'(x) = f(x)g'(x) = 4x^(4x)(1+lnx)#
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Answer 2

#dy/dx=4x^(4x)(lnx+1)#

Take the natural log of both sides.

#lny=lnx^(4x)#

#loga^b=bloga#. So,

#lny=4xlnx#

Take the derivative of both sides.

#(1/y)dy/dx=4+4lnx#

Multiply both sides by #y# and substitute in the original equation.

#dy/dx=4x^(4x)(lnx+1)#

I'm not here to judge, but I noticed this is the second logarithmic differentiation problem I have answered for you. Perhaps, you should try to better understand logarithmic differentiation. It is actually quite simple.

Here is a helpful video if you are interested in learning more.

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Answer 3
Given: #y = x^(4x)#

Use the natural logarithm on both sides:

#ln(y) = ln(x^(4x))#
Use the property of logarithms #ln(x^(4x)) = (4x)ln(x)#:
#ln(y) = (4x)ln(x)#

Differentiate both sides:

#(d(ln(y)))/dx = (d((4x)ln(x)))/dx#

The left side requires the use of the chain rule:

#1/ydy/dx = (d((4x)ln(x)))/dx#

The right side requires the use of the product rule:

#1/ydy/dx = (d(4x))/dxln(x) + 4x(d(ln(x)))/dx#
For the first term we use #(d(4x))/dx = 4#
#1/ydy/dx = 4ln(x) + 4x(d(ln(x)))/dx#
For the second term we use #(d(ln(x)))/dx = 1/x#
#1/ydy/dx = 4ln(x) + 4x1/x#

The second term simplifies:

#1/ydy/dx = 4ln(x) + 4#

Multiply both sides by y:

#dy/dx = (4ln(x) + 4)y#
Substitute #x^(4x)# for y:
#dy/dx = (4ln(x) + 4)x^(4x)#
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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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