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 ) = d d x [ ∫ 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)={\frac {d}{dx}}\left[\int _{0}^{tan(x)}{\sqrt {t+{\sqrt {t}}}}\,dt\right]=\sec ^{2}(x)\cdot {\sqrt {tan(x)+{\sqrt {tan(x)}}}})-0\cdot {\sqrt {0+{\sqrt {0}}}}\,=\sec ^{2}(x)\cdot {\sqrt {tan(x)+{\sqrt {tan(x)}}}})\end{aligned}}}
Therefore, y ′ = sec 2 ( x ) ⋅ t a n ( x ) + t a n ( x ) ) {\displaystyle {\text{Therefore, }}y'=\sec ^{2}(x)\cdot {\sqrt {tan(x)+{\sqrt {tan(x)}}}})}
![{\displaystyle {\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 {tan(x)}}}})-0\cdot {\sqrt {0+{\sqrt {0}}}}\,=\sec ^{2}(x)\cdot {\sqrt {tan(x)+{\sqrt {tan(x)}}}})\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/dc58a595f2aad3e2c34089e3ccadae24a38c442f)
