Question

find f(x), f(x), and f^{(3)}(x) for the following function. f(x)=7x^{3}+9x^{2}+2x f(x)=□ f(x)=□ f^{(3)}(x)=□

Explanation:

Step1: Apply power - rule for first - derivative

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $f(x)=7x^{3}+9x^{2}+2x$, we have $f^\prime(x)=7\times3x^{2}+9\times2x + 2=21x^{2}+18x + 2$.

Step2: Apply power - rule for second - derivative

Differentiate $f^\prime(x)=21x^{2}+18x + 2$ using the power - rule. So $f^{\prime\prime}(x)=21\times2x+18=42x + 18$.

Step3: Apply power - rule for third - derivative

Differentiate $f^{\prime\prime}(x)=42x + 18$ using the power - rule. Then $f^{(3)}(x)=42$.

Answer:

$f^\prime(x)=21x^{2}+18x + 2$
$f^{\prime\prime}(x)=42x + 18$
$f^{(3)}(x)=42$