Question

What is the derivative lime of y=mx+b?

Answer 1

`dy/dx=m`.

Explanation:

This can be shown two ways. First, using the limit definition of the derivative, we have
`dy/dx=lim_(c->0)(m(x+c)+b-(mx+b))/c`
`dy/dx=lim_(c->0)(mx+mc+b-mx-b)/c`
`dy/dx=lim_(c->0)m+b/c`
`dy/dx=m`
Alternatively, we can think of this geometrically. The derivative is defined as the slope of the tangent line at any point along the line `y=mx+b`. We know that the slope of the line `y=mx+b` is just `m`, so the slope of the tangent line along any point on the line `y=mx+b` is just `m`. Therefore, `dy/dx=m`.

Answer 2

`y'=m`

Explanation:

Given: `y=mx+b`.

where:

• `m` is the slope of the line

• `b` is the y-intercept

We see that `b` is a constant, so differentiating, we need two rules:

• `dy/dx(C)=0`

• `dy/dx(ax)=a`

Therefore, the derivative of this line is:

`y'=m`