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What is the derivative of # f(x) = 3x^2 ln 2x#?

Answer 1

Luke Alvarez

#d/dx ( 3x^2ln2x) = 3x( 1+ 2ln 2x )#

Applying the rule of the product:

#d/dx ( 3x^2ln2x) = d/dx(3x^2)ln 2x + 3x^2 d/dx (ln2x)#

#d/dx ( 3x^2ln2x) = 6xln 2x + 3x^2 2/(2x)#

#d/dx ( 3x^2ln2x) = 6xln 2x + 3x#

#d/dx ( 3x^2ln2x) = 3x( 1+ 2ln 2x )#

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Answer 2

Axel Alvarez

To find the derivative of ( f(x) = 3x^2 \ln(2x) ), you can use the product rule and the chain rule.

First, apply the product rule:

[ f'(x) = (3x^2)' \cdot \ln(2x) + 3x^2 \cdot (\ln(2x))' ]

Now, differentiate each part:

[ (3x^2)' = 6x ]

[ (\ln(2x))' = \frac{1}{2x} \cdot 2 = \frac{1}{x} ]

Substitute these derivatives back into the equation:

[ f'(x) = 6x \ln(2x) + 3x^2 \cdot \frac{1}{x} ]

Simplify:

[ f'(x) = 6x \ln(2x) + 3 ]

So, the derivative of ( f(x) = 3x^2 \ln(2x) ) is ( f'(x) = 6x \ln(2x) + 3 ).

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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.