Question

Differentiate underoot sec x

Answer 1

$\tt \longrightarrow \dfrac{d( \sqrt{sec \: x} )}{dx} \\ \\ \tt \longrightarrow \dfrac{d{(sec \: x)}^{ \dfrac{1}{2} } }{dx} \\ \\ \tt \longrightarrow \dfrac{1}{2} \times (sec \: x)^{1 - \frac{1}{2} } \times \dfrac{d{(sec \: x)}^{ } }{dx} \times \dfrac{dx}{dx} \\ \\ \tt \longrightarrow \frac{1}{2} \times (sec \: x)^{- \dfrac{1}{2} } {(sec \: x \times \: tan \: x)} \\ \\ \tt \longrightarrow \frac{1}{2} \times \dfrac{{(sec \: x \times \: tan \: x)}}{(sec \: x)^{ \frac{1}{2} }} \\ \\ \tt \longrightarrow \frac{{(sec \: x \times \: tan \: x)}}{2 \: \sqrt{sec \: x} } \\ \\ \tt \longrightarrow \frac{{(sec \: x \: \: tan \: x)}}{2 \: \sqrt{sec \: x} }$

Learn more :

d(e^x)/dx = e^x

d(x^n)/dx = n x^(n-1)

d(ln x)/dx = 1/x

d(sin x)/dx = cos x

d(cos x)/dx = - sin x

d(tan x)/dx = sec² x

d(sec x)/dx = tan x * sec x

d(cot x)/dx = - cosec²x

d(cosec x)/dx = - cosec x * cot x

Answer 2

${\bold{\sf{\underline{Answer}}}}$

$\: \:{\bold{\bf{\leadsto\dfrac{d(\sqrt{sec \: x})}{dx}}}}$

$\: \:{\bold{\bf{\leadsto \dfrac{d \: (sec \: x)\dfrac{1}{2}}{dx}}}}$

$\: \:{\bold{\bf{\leadsto \dfrac{1}{2} \times {(sec \: x)}^{1- \dfrac{1}{2}} \times \dfrac{d \: sec \: x}{dx} \times \dfrac{dx}{dx}}}}$

$\: \:{\bold{\bf{\leadsto \dfrac{1}{2} \times \: {(sec \: x)}^{\dfrac{-1}{2}} ( sec \: x \times \: tan \: x )}}}$

$\: \:{\bold{\bf{\leadsto \dfrac{1}{2} \times \: \dfrac{sec \: x \times \: tan \:x}{(sec \: x)^\dfrac{1}{2}}}}}$

$\: \:{\bold{\bf{\leadsto \dfrac{sec \: x \times \: tan \: x}{2 \sqrt{sec \: x}}}}}$

$\: \:{\bold{\bf{\leadsto \dfrac{sec \: x \: tan \: x}{2 \sqrt{sec \: x}}}}}$

${\bold{\sf{\underline{More \: knowledge}}}}$

Trigonometric Full Table

$\Large{ \begin{tabular}{|c|c|c|c|c|c|} \cline{1-6} \theta & \sf 0^{\circ} & \sf 30^{\circ} & \sf 45^{\circ} & \sf 60^{\circ} & \sf 90^{\circ} \\ \cline{1-6} \sin & 0 & \dfrac{1}{2 } & \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} & 1 \\ \cline{1-6} \cos & 1 & \dfrac{ \sqrt{ 3 }}{2} } & \dfrac{1}{ \sqrt{2} } & \dfrac{ 1 }{ 2 } & 0 \\ \cline{1-6} \tan & 0 & \dfrac{1}{ \sqrt{3} } & 1 & \sqrt{3} & \infty \\ \cline{1-6} \cot & \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } &0 \\ \cline{1 - 6} \sec & 1 & \dfrac{2}{ \sqrt{3}} & \sqrt{2} & 2 & \infty \\ \cline{1-6} \csc & \infty & 2 & \sqrt{2 } & \dfrac{ 2 }{ \sqrt{ 3 } } & 1 \\ \cline{1 - 6}\end{tabular}}$

Trigonometric Table

$\bullet\:\sf Trigonometric\:Values :\\\\\boxed{\begin{tabular}{c|c|c|c|c|c}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}&Not D{e}fined\end{tabular}}$