How do you find the first derivative of #y=(lnx)^tanx#?
Take the natural logarithm of both sides.
Differentiate both sides.
By the quotient rule:
Next, apply the product rule:
I know it's quite messy, but it worked!!
Practice exercises:
Hopefully this helps, and good luck!
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To find the first derivative of ( y = (\ln x)^{\tan x} ), we use the chain rule. The chain rule states that if we have a composite function ( f(g(x)) ), then the derivative is ( f'(g(x)) \cdot g'(x) ).
- Let ( u = \ln x ) and ( v = \tan x ).
- Rewrite ( y ) as ( y = u^v ).
- Find the derivatives ( \frac{du}{dx} ) and ( \frac{dv}{dx} ).
- Apply the chain rule: ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} + \frac{dy}{dv} \cdot \frac{dv}{dx} ).
- Substitute ( \frac{dy}{du} ) and ( \frac{dy}{dv} ).
- Simplify the expression to find ( \frac{dy}{dx} ).
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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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