# How do you find the derivatives of #y=(sintheta)^tantheta# by logarithmic differentiation?

usinf logarithmic differentiation

by laws of logs

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To find the derivative of ( y = (\sin(\theta))^{\tan(\theta)} ) using logarithmic differentiation:

- Take the natural logarithm of both sides: ( \ln(y) = \ln\left((\sin(\theta))^{\tan(\theta)}\right) ).
- Use the properties of logarithms to simplify the expression: ( \ln(y) = \tan(\theta) \cdot \ln(\sin(\theta)) ).
- Differentiate both sides with respect to ( \theta ).
- Apply the chain rule and product rule as necessary.
- Solve for ( \frac{dy}{d\theta} ) to find the derivative.

The derivative can be expressed as ( \frac{dy}{d\theta} = \frac{\cos(\theta) \ln(\sin(\theta)) + \frac{\tan(\theta)}{\sin(\theta)}}{(\sin(\theta))^{\tan(\theta)}} ).

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