find f(x), f(x), and f^(3)(x) for the following function. f(x)=8x^3 + 9x^2 + 4x f(x)=□

Answer:

$f^{\prime}(x)=24x^{2}+18x + 4$ $f^{\prime\prime}(x)=48x+18$ $f^{(3)}(x)=48$

Explanation:

Step1: Apply power - rule for first - derivative

If $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. For $f(x)=8x^{3}+9x^{2}+4x$, $f^{\prime}(x)=8\times3x^{2}+9\times2x + 4\times1=24x^{2}+18x + 4$.

Step2: Apply power - rule for second - derivative

Differentiate $f^{\prime}(x)=24x^{2}+18x + 4$. $f^{\prime\prime}(x)=24\times2x+18\times1=48x + 18$.

Step3: Apply power - rule for third - derivative

Differentiate $f^{\prime\prime}(x)=48x + 18$. $f^{(3)}(x)=48\times1+0 = 48$.