Question

Find the derivative of f(x) = 4 divided by x at x = 2.

Answer 1

`f'(2) = -1`

Step-by-step explanation

The given function is

`f(x) = (4)/(x)`

Recall that:

`\frac{c}{ {a}^( m) } = c {a}^( - m)`

We rewrite the given function using this rule to obtain,

`f(x) = 4 {x}^( - 1)`

Recall again that,

If

`f(x)= a {x}^(n)`

then

`f'(x)=n a {x}^(n - 1)`

We differentiate using the power rule to obtain,

`f'(x) = - 1 * 4 {x}^( - 1 - 1)`

`f'(x) = - 4 {x}^( - 2)`

We rewrite as positive index to obtain,

`f'(x) = - \frac{4}{ {x}^(2) }`

We plug in x=2 to obtain,

`f'(2) = - \frac{4}{ { (2)}^(2) } = - (4)/(4) = - 1`

Answer 2

Hello!

The answer is:

`f'(2)=-1`

Why?

To solve this problem, first we need to derivate the given function, and then, evaluate the derivated function with x equal to 2.

The given function is:

`f(x)=(4)/(x)`

It's a quotient, so, we need to use the following formula to derivate it:

`f'(x)=(d)/(dx)((u)/(v)) =(v*u'-u*v')/(v^(2) )`

Then, of the given function we have that:

`u=4\\v=x`

So, derivating we have:

`f'(x)=(d)/(dx)((4)/(x)) =(x*(4)'-4*(x)')/(x^(2) )`

`f'(x)=(d)/(dx)((4)/(x)) =(x*0-4*1)/(x^(2) )`

`f'(x)=(d)/(dx)((4)/(x)) =(0-4)/(x^(2) )`

`f'(x)=(d)/(dx)((4)/(x)) =(-4)/(x^(2) )`

Hence,

`f'(x)=(d)/(dx)((4)/(x)) =(-4)/(x^(2) )`

Now, evaluating with x equal to 2, we have:

`f'(2)=(-4)/((2)^(2) )`

`f'(2)=(-4)/(4)`

`f'(2)=-1`

Therefore, the answer is:

`f'(2)=-1`

Have a nice day!