How do you differentiate #y=(sinx)^lnx#?
# dy/dx = (sinx)^(lnx){(lnx)(cotx) + (ln(sinx))/x} #
Generally when dealing with a variable exponent it is easier to differentiate (and understand) by taking natural logarithms (to remove the exponent) and differentiating implicitly:
We have:
Take Natural logarithms:
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To differentiate ( y = (\sin x)^{\ln x} ), you would use the chain rule along with logarithmic differentiation. Here's the stepbystep process:

Take the natural logarithm of both sides of the equation to simplify the function. [ \ln y = \ln \left( (\sin x)^{\ln x} \right) ]

Apply the logarithm property (\ln(a^b) = b \ln(a)) to simplify the expression. [ \ln y = \ln x \cdot \ln(\sin x) ]

Differentiate both sides of the equation with respect to (x). [ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{1}{x} \cdot \ln(\sin x) + \frac{\ln x}{\sin x \cdot \cos x} ]

Solve for ( \frac{dy}{dx} ). [ \frac{dy}{dx} = y \cdot \left( \frac{1}{x} \cdot \ln(\sin x) + \frac{\ln x}{\sin x \cdot \cos x} \right) ]

Substitute back the original function (y = (\sin x)^{\ln x}) to get the final result. [ \frac{dy}{dx} = (\sin x)^{\ln x} \left( \frac{1}{x} \cdot \ln(\sin x) + \frac{\ln x}{\sin x \cdot \cos x} \right) ]
That's the derivative of (y) with respect to (x).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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