Question

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.66. y = θsin(θ)/√(sec(θ))

Accepted Answer

Start by rewriting the function to make logarithmic differentiation easier. Given $$ y = \frac{	heta \sin(	heta)}{\sqrt{\sec(	heta)}} $$, express the denominator with a fractional exponent: $$ y = 	heta \sin(	heta) \cdot (\sec(	heta))^{-1/2} . $$Take the natural logarithm of both sides to apply logarithmic differentiation: $$ \ln y = \ln 	heta + \ln \sin(	heta) + \ln (\sec(	heta))^{-1/2} . $$Simplify the logarithm of the power term using the property $$ \ln(a^b) = b \ln a $$: $$ \ln y = \ln 	heta + \ln \sin(	heta) - \frac{1}{2} \ln \sec(	heta) . $$Differentiate both sides with respect to $$ 	heta $$, remembering that $$ \frac{d}{d	heta} \ln y = \frac{1}{y} \frac{dy}{d	heta}  by $$the chain rule. So, $$ \frac{1}{y} \frac{dy}{d	heta} = \frac{1}{	heta} + \cot(	heta) - \frac{1}{2} \cdot \frac{d}{d	heta} \ln \sec(	heta) . $$Find the derivative of $$ \ln \sec(	heta) $$ using the chain rule: $$ \frac{d}{d	heta} \ln \sec(	heta) = 	an(	heta) . $$Substitute this back into the equation and then solve for $$ \frac{dy}{d	heta}  by $$multiplying both sides by $$ y .$$

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
66. y = θsin(θ)/√(sec(θ))

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Key Concepts

Logarithmic Differentiation

Properties of Logarithms

Derivative of Trigonometric Functions