 Derivatives of Functions of f(x) = u

 Introduction

Considering f(x) = (x + x2)5(1 + x3)2, find f'(x)

Letĺs re-write the function as follows:

`y = (x2 + x)5(1 + x3)2`

Letĺs consider a function k as follows:

```k  =  x2 + x
k' = 2x  + 1```

Letĺs consider a function h as follows:

```u  = k5
= (x2 + x)5

u' = nkn-1k'
= 5(x2 + x)4(2x + 1)
= 5(2x + 1)(x2 + x)4```

Letĺs consider a function g as follows:

```g  = 1 + x3

gĺ = 3x2```

Letĺs consider a function m as follows:

```v  = g2
= (1 + x3)2

v' = ngn-1g'
= 2(1 + x3)(3x2)
= 6x2(1 + x3)```

Letĺs consider a function f as follows:

`f  = uv`

To calculate the derivative, we use the formula (uv)' = u'v + uv':

```f' = u'v + uv'
= 5(2x + 1)(x2 + x)4(1 + x3)2 +(x2 + x)56x2(1 + x3)
= 5(2x + 1)(x2 + x)4(1 + x3)2 +6x2(1 + x3)(x2 + x)5
= (x2 + x)4(1 + x3)[5(2x + 1)(1 + x3) + 6x2(x2 + x)]```

 Considering f(x) = , find f'(x)

Letĺs consider a function k as follows:

```g  = 3x3 + 5
g' = 9x2 ```

Letĺs consider a function h as follows:

```u  = g2
= (3x3 + 5)2

u' = ngn-1g'
= 3(3x3 + 5)(9x2)
= 27x2(3x3 + 5)```

Letĺs consider a function k as follows:

```k  = 4x2 + 2x + 3

kĺ = 8x + 2```

Letĺs consider a function h as follows:

```v  = k4
= (4x2 + 2x + 3)4

vĺ = nkn-1k'
= 4(4x2 + 2x + 3)3(8x + 2)
= 4(8x + 2)(4x2 + 2x + 3)3```

Letĺs consider a function f as follows:

```f  = u/v
= (3x3 + 5)2/(4x2 + 2x + 3)4

f' = (u'v + uv')/v8 ```  Considering f(x) = [3x + (2x + x5)-2]-4, find f'(x)

Letĺs re-write the function as follows:

`y = (3x + (x5 + 2x)-2)-4`

Letĺs consider a function k as follows:

```k  =  x5 + 2x
k' = 5x4 + 2```

Letĺs consider a function h as follows:

```h  = k-2
= (x5 + 2x)-2

h' = nkn-1k'
= -2( x5 + 2x)-3(5x4 + 2)
= -2(5x4 + 2)(x5 + 2x)-3```

Letĺs consider a function g as follows:

```g  = 3x + h
= 3x + (x5 + 2x)-2

gĺ = (3x)' + h'
=  3    + (-2(5x4 + 2)(x5 + 2x)-3)
= -2(5x4 + 2)(x5 + 2x)-3 + 3```

Letĺs consider a function f as follows:

```f  = g-4
= [3x + (x5 + 2x)-2]-4

f' = ngn-1g'
= -4[3x + (x5 + 2x)-2]-5[-2(5x4 + 2)(x5 + 2x)-3 + 3]
= -4[-2(5x4 + 2)(x5 + 2x)-3 + 3][3x + (x5 + 2x)-2]-5```