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How do you find the derivative of #y=(lnx)^n#?

Answer 1

Samantha Adkison

#dy/dx =(n (ln x)^(n-1))/x#

Using the chain rule:

#dy/dx = d/dx (lnx)^n = n (ln x)^(n-1) d/dx (lnx) = (n (ln x)^(n-1))/x#

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Answer 2

Scarlett Allison

To find the derivative of ( y = (\ln x)^n ), where ( n ) is a constant:

Use the chain rule, which states that if ( u ) is a differentiable function of ( x ) and ( n ) is a constant, then the derivative of ( u^n ) with respect to ( x ) is ( n \cdot u^{(n-1)} \cdot \frac{du}{dx} ).
Let ( u = \ln x ).
Find ( \frac{du}{dx} ), which is the derivative of ( \ln x ) with respect to ( x ), equal to ( \frac{1}{x} ).
Apply the chain rule to ( u^n ), yielding ( n \cdot (\ln x)^{(n-1)} \cdot \frac{1}{x} ).
Substitute back ( u = \ln x ), resulting in ( \frac{n \cdot (\ln x)^{(n-1)}}{x} ).

So, the derivative of ( y = (\ln x)^n ) with respect to ( x ) is ( \frac{n \cdot (\ln x)^{(n-1)}}{x} ).

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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.