Question

What is the derivative of `-5x`?

Answer 1

`-5`

Explanation:

now the power rule for differentiation is:

`d/(dx)(ax^n)=anx^(n-1)`

`:.d/(dx)(-5x)`

`=d/(dx)(-5x^1)`

`=-5xx1xx x^(1-1)`

using the power rule

`=-5x^0=-5`

if we use the definition

`(dy)/(dx)=Lim_(h rarr0)(f(x+h)-f(x))/h`

we have

`(dy)/(dx)=Lim_(h rarr0)(-5(x+h)- -5x)/h`

`(dy)/(dx)=Lim_(h rarr0)(-5x-5h+5x)/h`

`(dy)/(dx)=Lim_(h rarr0)(-5h)/h`

`(dy)/(dx)=Lim_(h rarr0)(-5)=-5`

as before

Answer 2

-5

Explanation:

We can say
`f(x)=-5x`
The derivative of `f(x)` is defined as

`lim_(h->0)(f(x+h)-f(x))/h`

So,

`"The Derivative of f(x)"=lim_(h->0)(-5x-5h-(-5x))/h`

`=lim_(h->0)(-5x+5x-5h)/h`

`=lim_(h->0)(-5h)/h`

`=-5`

Hope it'd help.