y = ∫ 0 t a n ( x ) t + t d t {\displaystyle y=\int _{0}^{tan(x)}{\sqrt {t+{\sqrt {t}}}}\,dt}
d d x ( y ) = ∫ 0 t a n ( x ) t + t d t = sec 2 ( x ) ⋅ t a n ( x ) + t a n ( x ) ) − 0 ⋅ 0 + 0 = sec 2 ( x ) ⋅ t a n ( x ) + t a n ( x ) ) {\displaystyle {\begin{aligned}{\frac {d}{dx}}(y)=\int _{0}^{tan(x)}{\sqrt {t+{\sqrt {t}}}}\,dt\,=\sec ^{2}(x)\cdot {\sqrt {tan(x)+{\sqrt {t}}an(x)}})-0\cdot {\sqrt {0+{\sqrt {0}}}}\,=\sec ^{2}(x)\cdot {\sqrt {tan(x)+{\sqrt {t}}an(x)}})\end{aligned}}}
In this problem a ′ ( x ) = 0 {\displaystyle a^{\prime }{(x)}=0} , so when it is multiplied by f ( a ( x ) ) {\displaystyle f(a(x))} it will result in 0 and doesn't need to be added.
, so when it is multiplied by 
it will result in 0 and doesn't need to be added.