Question

(a) if $f(x)=x^{4}+6x$, find $f(x)$. $f(x)=$ (b) check to see that your answer to part (a) is reasonable by comparing the graphs of $f$ and $f$. (a graphing calculator is recommended.)

Explanation:

Step1: Apply power - rule for differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$, and for a sum of functions $y = u+v$, $y^\prime=u^\prime + v^\prime$. Given $f(x)=x^{4}+6x$. For the first term $u = x^{4}$, by the power - rule $u^\prime=4x^{4 - 1}=4x^{3}$. For the second term $v = 6x$, using the power - rule (since $x^1$), $v^\prime=6\times1\times x^{1 - 1}=6$.

Step2: Find the derivative of $f(x)$

$f^\prime(x)=u^\prime + v^\prime=4x^{3}+6$.

Answer:

$4x^{3}+6$