5.3 The Fundamental Theorem of Calculus/15

$y=\int _{0}^{tan(x)}{\sqrt {t+{\sqrt {t}}}}\,dt$

${\begin{aligned}{\frac {d}{dx}}(y)={\frac {d}{dx}}\left[\int _{0}^{tan(x)}{\sqrt {t+{\sqrt {t}}}}\,dt\right]=\sec ^{2}(x)\cdot {\sqrt {tan(x)+{\sqrt {t}}an(x)}})-0\cdot {\sqrt {0+{\sqrt {0}}}}\,=\sec ^{2}(x)\cdot {\sqrt {tan(x)+{\sqrt {t}}an(x)}})\end{aligned}}$

In this problem $a^{\prime }{(x)}=0$, so when it is multiplied by $f(a(x))$ it will result in 0 and doesn't need to be added.