Question

What is the derivative of the line \[y = mx + b\] ?

Answer 1

Hint: 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,
\[\dfrac{{dy}}{{dz}} = \dfrac{{mdx}}{{dz}}\].