Question

What is the derivative graph of a parabola?

Answer 1

Hint: In this problem we need to find the derivative graph of a parabola. For this we will first assume the standard equation of the parabola which is given by $y=a{x}^2+bx+c$ where $a$, $b$, $c$ are the constants. Now we will differentiate the above equation with respect to the variable $x$. By using the differentiation formula, we will simplify the obtained equation to get the required result.

Complete step-by-step answer:
Let the equation of the parabola will be $y=a{x}^2+bx+c$.
Differentiating the above equation with respect to $x$, then we will get
${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{d}{dx}}(a{x}^2+bx+c)$
Applying the differentiation for each term individually, then we will have
${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{d}{dx}}\left(a{x}^2\right)+{\displaystyle \frac{d}{dx}}\left(bx\right)+{\displaystyle \frac{d}{dx}}\left(c\right)$
Taking out the constants from differentiation which are in multiplication with the variables in the above equation, then we will get
${\displaystyle \frac{dy}{dx}}=a{\displaystyle \frac{d}{dx}}\left(x^2\right)+b{\displaystyle \frac{d}{dx}}\left(x\right)+{\displaystyle \frac{d}{dx}}\left(c\right)$
From the differentiation formula ${\displaystyle \frac{d}{dx}}\left(x^n\right)=n{x}^{n-1}$ we can write the value of ${\displaystyle \frac{d}{dx}}\left(x^2\right)$ as $2x$. Substituting this value in the above equation, then we will have
${\displaystyle \frac{dy}{dx}}=a\left(2x\right)+b{\displaystyle \frac{d}{dx}}\left(x\right)+{\displaystyle \frac{d}{dx}}\left(c\right)$
We have the value ${\displaystyle \frac{d}{dx}}\left(x\right)=1$. Substituting this value in the above equation, then we will get
${\displaystyle \frac{dy}{dx}}=a\left(2x\right)+b\left(1\right)+{\displaystyle \frac{d}{dx}}\left(c\right)$
The value $c$ which is in the above equation is a constant. We know that differentiation value of a constant is equals to zero. By using this value, we can write the above equation as
${\displaystyle \frac{dy}{dx}}=a\left(2x\right)+b\left(1\right)+0$
Simplifying the above equation by using the basic mathematical operations, then we will get
${\displaystyle \frac{dy}{dx}}=2ax+b$
The equation $2ax+b$ represents the equation of the line. We can observe this in the below graph also

Hence the derivative graph of the parabola is Straight Line.

Note: In this problem we have assumed the equation of the parabola as $y=a{x}^2+bx+c$ instead of $x=a{y}^2+by+c$ even though the both the equations represent the parabola. Because the equation $y=a{x}^2+bx+c$ has an extra advantage to calculate the differentiation value easily over the equation $x=a{y}^2+by+c$.