Question

How do you find the first and second derivative of ${x^2} - 2x - 3$?

Answer 1

Let $f(x) = {x^2} - 2x - 3$. Then $f'(x)$ and $f''(x)$ are the first and second derivatives respectively. The derivative of a function is the change in the value of the function with respect to the small change in the value of $x$, i.e. $\dfrac{{df(x)}}{{dx}}$. In this question we will find the derivatives using standard forms.

Complete step by step solution:
The given function is ${x^2} - 2x - 3$.
Let $f(x) = {x^2} - 2x - 3$
Then we have to find the first derivative, i.e. $\dfrac{{df(x)}}{{dx}}$, and the second derivative, i.e. $\dfrac{{{d^2}f(x)}}{{d{x^2}}}$.
The first derivative is $f'(x) = \dfrac{{df(x)}}{{dx}} = \dfrac{{d({x^2} - 2x - 3)}}{{dx}}$
We have, if $f(x) = g(x) + h(x)$, then $\dfrac{{df(x)}}{{dx}} = \dfrac{{dg(x)}}{{dx}} + \dfrac{{dh(x)}}{{dx}}$, i.e. we can find the derivative of each term separately.
Thus,
$f'(x) = \dfrac{{d({x^2} - 2x - 3)}}{{dx}} = \dfrac{{d({x^2})}}{{dx}} + \dfrac{{d( - 2x)}}{{dx}} + \dfrac{{d( - 3)}}{{dx}}$
Also we know,
$\dfrac{{d({x^n})}}{{dx}} = n{x^{n - 1}}$
$\dfrac{{dcf(x)}}{{dx}} = c\dfrac{{df(x)}}{{dx}}$ , where $c$ is any constant number
$\dfrac{{dc}}{{dx}} = 0$, where $c$ is any constant number
Thus, we can write the first derivative as,

$$\dfrac{{df(x)}}{{dx}} = \dfrac{{d({x^2})}}{{dx}} + \dfrac{{d( - 2x)}}{{dx}} + \dfrac{{d( - 3)}}{{dx}} \\\ \Rightarrow \dfrac{{df(x)}}{{dx}} = 2x - 2\dfrac{{dx}}{{dx}} + 0 = 2x - 2 \\\$$

Thus, the first derivative of the given function is $f'(x) = 2x - 2$
Now we have to further differentiate the function that we got as the first derivative to find the second derivative.
$f''(x) = \dfrac{{df'(x)}}{{dx}} = \dfrac{{{d^2}f(x)}}{{d{x^2}}} \\\ \Rightarrow f''(x) = \dfrac{{d(2x - 2)}}{{dx}} \\\ \Rightarrow f''(x) = \dfrac{{d(2x)}}{{dx}} + \dfrac{{d( - 2)}}{{dx}} \\\ \Rightarrow f''(x) = 2\dfrac{{dx}}{{dx}} + 0 = 2 \\\$
Thus, the second derivative of the given function is $f''(x) = 2$
Hence, we got the first derivative of the given function as $f'(x) = 2x - 2$ and the second derivative as $f''(x) = 2$.

Note:
The derivative of a function is the change in the value of the function for a very small change in the value of the variable. The derivative of a function can itself be a function which can be further differentiated. The second derivative is the derivative of the first derivative. We used standard forms of differentiation to find the derivatives. One other way is to find the derivative using first principles.