Question

How do you find the derivative of `g(x) = -4x + 5`?

Answer 1

`g'(x)=-4`

Explanation:

There are two ways to go about this. The first, I will be using the rules of derivatives. The second, I'll be showing by first principles.

Method 1: Rules of Derivatives

There are two rules we'll be using to solve this. The first is the derivative of a polynomial, which says:

`d/dx(x^n)=nx^(n-1)`

`d/dx(n)=0`

The highest power we have is 1, so by bringing that down, we remove the x since `x^0=1`.

By applying both rules, we can get:

`d/dx(-4x+5)=-4`

Method 2: First Principles

This method can be used to take the derivative of literally any function. A derivative is the instantaneous rate of change at a point, and we want the function for it. So we'll apply the rate of change formula (`(rise)/(run)`) and get as close to 0 as possible.

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

So let's plug in our function:

`lim_(h->0)((-4(x+h)+5-(-4x+5))/h)`

Let's expand this.

`lim_(h->0)((-4x-4h+5+4x-5)/h)`

And let's get the like terms together:

`lim_(h->0)((-4h)/h)`

And as that approaches 0, we just get

`g'(x)=-4`

Now, you have two methods of finding the derivative. One for polynomials, one for any other function.