How do I find the derivative of #y = (sin(e^x))^ln(x^2)#?
The given equation is
Take the natural logarithm of both sides then differentiate with respect to x
We now have a derivative of a product at the right sides of the equation.
Obtain the derivative of both sides
God bless....I hope the explanation is useful.
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To find the derivative of (y = (\sin(e^x))^{\ln(x^2)}), you can use the chain rule and the properties of logarithmic and exponential functions. Here's the stepbystep process:

Take the natural logarithm of both sides of the equation: [ \ln(y) = \ln\left((\sin(e^x))^{\ln(x^2)}\right) ]

Use the property (\ln(a^b) = b\ln(a)) to simplify: [ \ln(y) = \ln(x^2)\ln(\sin(e^x)) ]

Differentiate both sides with respect to (x): [ \frac{1}{y} \frac{dy}{dx} = \frac{2}{x} \ln(\sin(e^x)) + \frac{x^2}{\sin(e^x)} \cdot \frac{\cos(e^x)e^x}{\sin(e^x)} ]

Solve for (\frac{dy}{dx}): [ \frac{dy}{dx} = y \left(\frac{2}{x} \ln(\sin(e^x)) + \frac{x^2}{\sin(e^x)} \cdot \frac{\cos(e^x)e^x}{\sin(e^x)}\right) ]

Substitute back (y = (\sin(e^x))^{\ln(x^2)}): [ \frac{dy}{dx} = (\sin(e^x))^{\ln(x^2)} \left(\frac{2}{x} \ln(\sin(e^x)) + \frac{x^2}{\sin(e^x)} \cdot \frac{\cos(e^x)e^x}{\sin(e^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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