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

1 Answer
Apr 10, 2018

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.