How do you find the derivative for #y=(sin(3x))^(ln(x))#?

Answer 1
Use logarithmic differentiation. Take the natural logarithm of both sides of #y=(sin(3x))^{ln(x)}# to get #ln(y)=ln((sin(3x))^{ln(x)})#. Then use the property #ln(a^{b})=bln(a)# (for #a>0#) to write #ln(y)=ln(x)ln(sin(3x))#.
Now differentiate both sides of this last equation with respect to #x#, keeping in mind that #y# is a function of #x# and using the Chain Rule and the Product Rule:
#1/y * dy/dx = 1/x * ln(sin(3x))+ln(x)* 1/(sin(3x)) * 3cos(3x).#
Simplifying a bit and multiplying both sides of this last equation by #y=(sin(3x))^{ln(x)}# helps us conclude that
#dy/dx =(sin(3x))^{ln(x)}*(\frac{ln(sin(3x))}{x}+3ln(x)*cot(3x)) #
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Answer 2

To find the derivative of (y = (\sin(3x))^{\ln(x)}), we'll use the chain rule and the properties of logarithmic and exponential functions. Here's the process:

  1. Let (u = \sin(3x)) and (v = \ln(x)).
  2. Apply the chain rule to find (\frac{du}{dx}) and (\frac{dv}{dx}).
  3. Use the formula for differentiating (f(g(x))) where (f) is a function of (g(x)).
  4. Substitute (u = \sin(3x)), (v = \ln(x)), (\frac{du}{dx}), and (\frac{dv}{dx}) into the formula.
  5. Compute the derivative ( \frac{dy}{dx} ) by multiplying ( \frac{dy}{du} ) and ( \frac{du}{dx} ), and then adding the product of ( \frac{dy}{dv} ) and ( \frac{dv}{dx} ).

Let's go through the steps to find ( \frac{dy}{dx} ):

  1. (u = \sin(3x)) implies ( \frac{du}{dx} = 3\cos(3x) ).
  2. (v = \ln(x)) implies ( \frac{dv}{dx} = \frac{1}{x} ).
  3. (y = u^v) implies ( \frac{dy}{du} = vu^{v-1} ) and ( \frac{dy}{dv} = u^v \ln(u) ).
  4. Substitute ( \frac{du}{dx} ), ( \frac{dv}{dx} ), ( \frac{dy}{du} ), and ( \frac{dy}{dv} ) into the differentiation formula.
  5. Compute ( \frac{dy}{dx} ) by multiplying and adding the appropriate terms.

So, the derivative ( \frac{dy}{dx} ) of ( y = (\sin(3x))^{\ln(x)} ) is:

[ \frac{dy}{dx} = \frac{du}{dx} vu^{v-1} + \frac{dv}{dx}u^v \ln(u) ]

[ = (3\cos(3x))(\ln(x) \sin(3x)^{\ln(x)-1}) + \frac{1}{x}(\sin(3x))^{\ln(x)} \ln(\sin(3x)) ]

This is the derivative of the given function.

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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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