If $F ( z ) = \frac { \tan 6 z } { \ln z }$ , then $F ^ { \prime } ( z ) = \frac { \frac { 6 } { \cos ^ { 2 } z } \ln z - ( \tan 6 z ) \frac { 1 } { z } } { ( \ln z ) ^ { 2 } }$ .

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The derivative of $F(z)$ is given by the quotient rule:$$F'(z) = \frac{(\ln z)\frac{d}{dz}[\tan 6z] - (\tan 6z)\frac{d}{dz}[\ln z]}{(\ln z)^2}$$Using the chain rule, we have:$$\frac{d}{dz}[\tan 6z] = 6\sec^2 6z$$and$$\frac{d}{dz}[\ln z] = \frac{1}{z}$$Substituting these derivatives into the quotient rule, we get:$$F'(z) = \frac{(\ln z)(6\sec^2 6z) - (\tan 6z)\left(\frac{1}{z}\right)}{(\ln z)^2}$$Simplifying, we get:$$F'(z) = \frac{6\sec^2 6z\ln z - \frac{\tan 6z}{z}}{(\ln z)^2}$$$$F'(z) = \frac{\frac{6}{\cos^2 6z}\ln z - \tan 6z\frac{1}{z}}{(\ln z)^2}$$

If F(z)=tan⁡6zln⁡zF ( z ) = \frac { \tan 6 z } { \ln z }F(z)=lnztan6z​

If $F ( z ) = \frac { \tan 6 z } { \ln z }$