How do you find the derivative of #y=ln(1-2x)^3#?

Answer 1

#y' = -6/(1-2x) #

#" "# #y# is composite of two functions #" lnx" " and (1 - 2x)^3 # #" "# Let #u(x)= lnx " and " v(x) = (1-2x)^3" # #" "# Then, #" y=u(v(x))" # #" "# Differentiating this function is determined by applying chain rule. #" "# #color(red)(y' = u'(v(x))xxu'(x))# #" "# #color(red)(u'(v(x)) = ?)# #" "# #u'(x) = 1/x# #" "# #u'(v(x))) = 1/(v(x))# #" "# #" "# #color(red)(u'(v(x)) = 1/(1-2x)^3)# #" "# #" "# #" "# #color(red)(v'(x) = ?)# #" "# #v'(x) = 3(1-2x)^2(1-2x)'# #" "# #v'(x) = 3(1-2x)^2(-2)# #" "# #color(red)(v'(x) = -6(1-2x)^2)# #" "# #" "# #color(red)(y' = u'(v(x))xxu'(x))# #" "# #y' = 1/(1-2x)^3 xx -6(1-2x)^2 # #" "# #y' = (-6(1-2x)^2)/(1-2x)^3 # #" "# #y' = -6/(1-2x) #
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Answer 2

To find the derivative of ( y = \ln(1 - 2x)^3 ), we'll use the chain rule. The derivative of ( \ln(u) ) with respect to ( x ) is ( \frac{1}{u} \cdot \frac{du}{dx} ).

So, applying the chain rule, the derivative of ( y ) with respect to ( x ) is:

[ \frac{d}{dx} \left( \ln(1 - 2x)^3 \right) = \frac{1}{(1 - 2x)^3} \cdot \frac{d}{dx}(1 - 2x)^3 ]

Now, we'll find ( \frac{d}{dx}(1 - 2x)^3 ) using the power rule, which states that the derivative of ( u^n ) with respect to ( x ) is ( n \cdot u^{n-1} \cdot \frac{du}{dx} ). Applying this to ( (1 - 2x)^3 ), we get:

[ \frac{d}{dx}(1 - 2x)^3 = 3 \cdot (1 - 2x)^2 \cdot \frac{d}{dx}(1 - 2x) ]

Now, finding ( \frac{d}{dx}(1 - 2x) ), which is simply ( -2 ), we substitute it back:

[ \frac{d}{dx}(1 - 2x)^3 = 3 \cdot (1 - 2x)^2 \cdot (-2) = -6(1 - 2x)^2 ]

Putting it all together:

[ \frac{d}{dx} \left( \ln(1 - 2x)^3 \right) = \frac{1}{(1 - 2x)^3} \cdot (-6(1 - 2x)^2) = -\frac{6(1 - 2x)^2}{(1 - 2x)^3} = -\frac{6}{1 - 2x} ]

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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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