Question

найдите производные функции:
1) f(x) = e^x/Inx
2) f(x) = x•e^x + 15
3) f(x) = 7^x/x^5
4) f(x) = 10x • e^x + 4x^2

Answer 1

Ответ:

Объяснение:

1) f(x) =u(x)/v(x) =>  f'(x)=  (u' (x)*v(x)-v '(x)*u(x))/ v²(x)

u(x)=$e^x v(x)=ln(x) = > u' (x) = e^x v'(x)= 1/x$

=>f(x)=($e^x$· lnx - $e^x$/x) /( ln x)² =(x·$e^x$·ln x - $e^x)$/(x· (ln x)²)

2) f(x) = u(x)·v(x)+ C    => f'(x) = u '(x)·v(x)+u(x)·v '(x) +0

u(x)= x   => u'(x)=1   v(x)= $e^x = > v' (x)= e^x$

=> f'(x) = $e^x+x*e^x$

3) f(x)= $7^x/x^5 = 7^x*x^-^5$

f(x)=u(x)·v(x) =>  f'(x) = u '(x)·v(x)+u(x)·v '(x)

u(x)=$7^x = > u'(x)= 7^x*ln7$    v(x)= x$^-^5 = > v'(x)= -5*x^-^6$

f'(x) = $7^x*lnx*x^-^5 -5*x^-^6*7^x$

4) f(x)= u(x)·v(x) +g(x) =>  f'(x) = u '(x)·v(x)+u(x)·v '(x) +g'(x)

u(x)=10x =>  u'(x)=10   v(x)=$e^x = > v'(x)=e^x$    g(x)=4x²     g'(x)=8x

f'(x) = 10·$e^x +10x*e^x +8x$