f ( x ) = e ( 2 x 2 ) + 1 {\displaystyle f(x)=e^{(2x^{2})+1}\,}
Erste Ableitung:
f ′ ( x ) = 4 ∗ x ∗ e ( 2 x 2 ) + 1 {\displaystyle f'(x)=4*x*e^{(2x^{2})+1}\,}
Zweite Ableitung:
g ( x ) = 4 ∗ x {\displaystyle g(x)=4*x\,}
h ( x ) = e ( 2 x 2 ) + 1 {\displaystyle h(x)=e^{(2x^{2})+1}\,}
f ″ ( x ) = g ′ ( x ) ∗ h ( x ) + g ( x ) ∗ h ′ ( x ) = 4 ∗ e ( 2 x 2 ) + 1 + 4 ∗ x ∗ 4 ∗ x ∗ e ( 2 x 2 ) + 1 {\displaystyle f''(x)=g'(x)*h(x)+g(x)*h'(x)=4*e^{(2x^{2})+1}+4*x*4*x*e^{(2x^{2})+1}\,}
f ( x ) = l n [ x + ( 1 + x 3 ) 1 / 2 ] {\displaystyle f(x)=ln{[x+(1+x^{3})^{1/2}]}\,}
g ( x ) = [ x + ( 1 + x 3 ) 1 / 2 ] {\displaystyle g(x)=[x+(1+x^{3})^{1/2}]\,}
h ( x ) = ( 1 + x 3 ) 1 / 2 {\displaystyle h(x)=(1+x^{3})^{1/2}\,}
h ′ ( x ) = 3 ∗ x 2 ∗ ( 1 + x 3 ) − 1 / 2 2 = 3 ∗ x 2 2 ∗ ( 1 + x 3 ) 1 / 2 {\displaystyle h'(x)={\frac {3*x^{2}*(1+x^{3})^{-1/2}}{2}}={\frac {3*x^{2}}{2*(1+x^{3})^{1/2}}}\,}
g ′ ( x ) = 1 + h ′ ( x ) {\displaystyle g'(x)=1+h'(x)\,}
f ′ ( x ) = g ′ ( x ) g ( x ) = 1 + 3 ∗ x 2 2 ∗ ( 1 + x 3 ) 1 / 2 [ x + ( 1 + x 3 ) 1 / 2 ] {\displaystyle f'(x)={\frac {g'(x)}{g(x)}}={\frac {1+{\frac {3*x^{2}}{2*(1+x^{3})^{1/2}}}}{[x+(1+x^{3})^{1/2}]}}\,}
![{\displaystyle f(x)=ln{[x+(1+x^{3})^{1/2}]}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1827a952d31203329793c31f857fc0d2e7d176)
![{\displaystyle g(x)=[x+(1+x^{3})^{1/2}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60e40fce5cc7a61ec9bf8806b5c2400f2b2189c4)
![{\displaystyle f'(x)={\frac {g'(x)}{g(x)}}={\frac {1+{\frac {3*x^{2}}{2*(1+x^{3})^{1/2}}}}{[x+(1+x^{3})^{1/2}]}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05bd72ee3debc0df6f0993287f0da823ce93f6b0)
