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How do you find the derivative of y=lnx3?

Answer
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Hint: Use the logarithm law lnab=blna and formula ddx(lnx)=1x then apply the formula of differentiation ddx(lnx)=1x

Complete step by step answer:
Here y=lnx3,
Using the logarithm law,
lnab=blna in y=lnx3 , where a is x and b is 3 .
Rewriting the equationy=lnx3 after applying logarithm law
y=3lnx
So, y=3lnx
Differentiating the above equation with respect to x
ddx3lnx
First, applying the differentiation formula d(kx)dx=k where k is constant .
3ddxlnx (3 is constant for ddx3lnx )
Now, applying the formula of differentiation, ddx(lnx)=1x
3×1x
3x

Thus, derivative of y=lnx3 is 3x.

Additional information:
In the graph of y=lnx3 the slope will be three times the slope of lnx.

Note: ddx(lnx)=1x can also be written as ddxlogx=1x.
The method of differentiating functions by first using logarithm laws to deduce and then differentiating is called logarithmic differentiation. We use logarithmic differentiation to make it easier to differentiate the logarithm of a function than to differentiate the function itself.
Whenever you come across logarithm questions, try to deduce them using logarithm law to simplify the log expression before differentiating it.
Logarithmic differentiation is used to transform products into sum and division into subtraction using the chain rule and properties of logarithms. The principle of logarithm differentiation can be used in some parts of the differentiation of differentiable functions, providing that these functions are non-zero.
When a base is not given a logarithm function you can automatically assume it as 10 and start solving.
Don’t assume y=lnx3 as y=(lnx)3 both are different functions . In this y=lnx3 x power’s is 3 and in this y=(lnx)3 whole power’s is 3.