Question

What is the derivative of the line $y = mx + b$ ?

Answer 1

We have a linear function $y = mx + b$, this function is an equation of linear equation of straight line. Here $m$ is the slope of the line and $b$ is the y-intercept of the line. So both $m$ and $b$ are the constants. We have to differentiate the function with respect to $x$ that is the derivative of the function.

Complete step-by-step solution:
Given,
A function $y = mx + b$ that is the equation of the straight line.
Here, $m$ is the slope of the line and
$b$ is the y-intercept of the line.
So both $m$ and $b$ are the constants.
To find,
Derivative of a linear function
$y = mx + b$ …………………………. (i)
On differentiating both side with respect to $x$
$\dfrac{{dy}}{{dx}} = \dfrac{{d(mx + b)}}{{dx}}$
Applying distributive property on derivatives.
Distributive property is $a(b + c) = ac + ab$
$\dfrac{{dy}}{{dx}} = \dfrac{{d(mx)}}{{dx}} + \dfrac{{d(b)}}{{dx}}$
Taking $m$outside of the derivative because $m$is the constant and the derivative of any constant term is $0$.
$\dfrac{{dy}}{{dx}} = m\dfrac{{dx}}{{dx}} + 0$
Derivative of $x$with respect to $x$is$1$.
$\dfrac{{dy}}{{dx}} = m(1)$ ……………………….( $\dfrac{{dx}}{{dx}} = 1$ )
$\dfrac{{dy}}{{dx}} = m$
Final answer:
Derivative of the function $y = mx + b$ is
$\Rightarrow \dfrac{{dy}}{{dx}} = m$

Note: Here, we have to use the concept of differentiation. In the particular question, we have to find the derivative of the function with respect to $x$ so the answer comes out looking good and in a good format. If they ask us to find the derivative of the function with respect to $z$then the answer does not come in this format that looks link this.
On differentiating both side with respect to $z$
$\dfrac{{dy}}{{dz}} = \dfrac{{d(mx)}}{{dz}} + \dfrac{{d(b)}}{{dz}}$
Derivative of any variable with another variable is written like $\dfrac{{dy}}{{dx}}$. Here, $y$is the first variable and we are differentiating with respect to $x$.
So, all those are written like this,
$\dfrac{{dy}}{{dz}} = \dfrac{{mdx}}{{dz}}$.