use logarithmic differentiation to find the derivative of the function y = (sin x)^(ln x)
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You need to take logarithms both sides such that:
`ln y = ln((sin x)^(ln x)) `
Using logarithmic identites yields:
`ln y = ln x*ln sin x `
You need to differentiate both sides with respect to x such that:
`(1/y)*y' = (ln x*ln sin x)'`
You should use product rule to the right side such that:
`(1/y)*y' = (ln x)'*ln sin x + (ln x)*(ln sin x)'`
`(1/y)*y' = (1/x)*ln sin x + (ln x)*(1/sin x)*(sin x)'`
`(1/y)*y' = (1/x)*ln sin x + (ln x)*(cos x/sin x)`
`(1/y)*y' = (1/x)*ln sin x + (ln x)*(cot x)`
`y' = y*((1/x)*ln sin x + (ln x)*(cot x))`
Substituting `(sin x)^(ln x) ` for y yields:
`y' = ((sin x)^(ln x))*((1/x)*ln sin x + (ln x)*(cot x)))`
Hence, differentiating the given function, using logarithmic differentiation, yields `y' = ((sin x)^(ln x))*((1/x)*ln sin x + (ln x)*(cot x))).`
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