Question

if $f(x)=5x^{2}-11x - 21$, find $f(a)$. answer:

Explanation:

Step1: Recall power - rule of differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$, and for a function $y=ax^n$ (where $a$ is a constant), $y^\prime = anx^{n - 1}$. Also, the derivative of a sum/difference of functions is the sum/difference of their derivatives, i.e., $(u\pm v\pm w)^\prime=u^\prime\pm v^\prime\pm w^\prime$.
For $f(x)=5x^{2}-11x - 21$, we find the derivative of each term separately.
The derivative of $5x^{2}$ using the power - rule: Let $a = 5$ and $n = 2$, then $(5x^{2})^\prime=5\times2x^{2 - 1}=10x$.
The derivative of $-11x$: Let $a=-11$ and $n = 1$, then $(-11x)^\prime=-11\times1x^{1 - 1}=-11$.
The derivative of the constant $-21$ is $0$ since the derivative of a constant $C$ is $C^\prime = 0$.
So, $f^\prime(x)=(5x^{2}-11x - 21)^\prime=(5x^{2})^\prime-(11x)^\prime-(21)^\prime=10x-11$.

Step2: Evaluate $f^\prime(x)$ at $x = a$

To find $f^\prime(a)$, we substitute $x = a$ into $f^\prime(x)$.
Since $f^\prime(x)=10x - 11$, then $f^\prime(a)=10a-11$.

Answer:

$10a - 11$