 Derivatives of f(x) = u

 Introduction

Considering f(x) = (2 - 5x)3, find f'(x)

Letĺs re-write the function as follows:

`y = (-5x + 2)3`

Let's consider the function

```u  = -5x + 2
u' = -5```

We know that the derivative of un, f'(u) = nun-1du/dx. Therefore:

```un    = (-5x + 2)3
(un)' = nun-1u'
= 3(-5x + 2)2(-5)
= -15(-5x + 2)2```

Considering f(x) = (7 - 2x3)-4, find f'(x)

Letĺs re-write the function as follows:

`y = (-2x3 + 7)-4`

Let's consider the function u as follows

```u  = -2x3 + 7
u' = -6x2```

Let's consider the function u as follows

`f  = u-4`

We know that the derivative of un, f'(u) = nun-1du/dx. Therefore:

```un    = (-2x3 + 7)-4
(un)' = nun-1u'
= -4(-2x3 + 7)-5(-6x2)
= -24x2(7 - 2x3)-5```
 Find the derivative of Let's consider

```u  = 4x - 7

u' = 4```

Remember the formula to find the derivative of the inverse of a function:

 `f'(x) = ` Find the derivative of Let's consider

```u  = x - 1

u' = 1```

And let's consider

```v  = x + 1

v' = 1```

Remember the formula to find the derivative of the division of two functions:

 ` `  ` `

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

Letĺs re-write the function as follows:

`y = (2x + 1)-3`

Let's consider the function

```u  = 2x + 1
u' = 2```

We know that the derivative of un, f'(u) = nun-1du/dx. Therefore:

```un    = (2x + 1)-3
(un)' = nun-1u'
= -3(2x + 1)-4(2)
= -6(2x + 1)-4```
 ` =` ` `

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