Question

How do you find the derivative of $f\left( x \right) = 2x - 1$?

Answer 1

Here, we will use the limit definition formula for the given functions. Then by substituting the limits, we will find the derivative of the function. Differentiation is a method of finding the derivative of the function and finding the rate of change of a function with respect to one variable.

Formula Used:
Limit definition is given by $f'\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{f\left( {x + \Delta x} \right) - f\left( x \right)}}{{\Delta x}}$ .

Complete Step by Step Solution:
We are given a function:
$f\left( x \right) = 2x - 1$ ………………………………………$\left( 1 \right)$
Now, we will find $f\left( {x + \Delta x} \right)$, so we get
$\Rightarrow f\left( {x + \Delta x} \right) = 2\left( {x + \Delta x} \right) - 1$ …………………………………………….$\left( 2 \right)$
Now, we will find the derivative of $2x - 1$ using the limit definition.
By substituting the equation $\left( 1 \right)$ and equation $\left( 2 \right)$ in the limit definition $f'\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{f\left( {x + \Delta x} \right) - f\left( x \right)}}{{\Delta x}}$, we get
$\Rightarrow f'\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{\left( {2\left( {x + \Delta x} \right) - 1} \right) - \left( {2x - 1} \right)}}{{\Delta x}}$
Multiplying the terms, we get
$\Rightarrow f'\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{2x + 2\Delta x - 1 - 2x + 1}}{{\Delta x}}$
Adding and subtracting the like terms, we get
$\Rightarrow f'\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \dfrac{{2\Delta x}}{{\Delta x}}$
Now, by simplifying the equation, we get
$\Rightarrow f'\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} 2$
Substituting the limits, we get
$\Rightarrow f'\left( x \right) = 2$

Therefore, the derivative of $2x - 1$ using the limit definition is $2$.

Additional information:
We know that the reverse process of differentiation is called antidifferentiation or integration. We should remember some rules in differentiation which include that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. The derivative of a constant is always zero since zero is a constant its derivative is zero.

Note:
We can also solve this question using an alternate method. We will find the derivative by using the differentiation formula.
Now differentiating both sides of the given function $f\left( x \right) = 2x - 1$ with respect to $x$, we get
$\Rightarrow f'\left( x \right) = \dfrac{d}{{dx}}\left( {2x} \right) - \dfrac{d}{{dx}}\left( 1 \right)$
Taking the constant out of differentiation, we get
$\Rightarrow f'\left( x \right) = 2\dfrac{d}{{dx}}\left( x \right) - \dfrac{d}{{dx}}\left( 1 \right)$
Now using the differentiation formula $\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}$ and $\dfrac{d}{{dx}}\left( k \right) = 0$, we get
$\Rightarrow f'\left( x \right) = 2 - 0$
$\Rightarrow f'\left( x \right) = 2$
Therefore, the derivative of $2x - 1$ is $2$.