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 of differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$, and for a sum of functions $u(x)+v(x)$, the derivative is $u^\prime(x)+v^\prime(x)$. For $f(x)=x^{4}+6x$, where $u(x)=x^{4}$ and $v(x)=6x$.
The derivative of $u(x)=x^{4}$ is $u^\prime(x) = 4x^{3}$ (using the power - rule with $n = 4$), and the derivative of $v(x)=6x$ is $v^\prime(x)=6$ (since for $y = ax$, $y^\prime=a$ with $a = 6$).

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

By the sum - rule of differentiation $f^\prime(x)=u^\prime(x)+v^\prime(x)$. So $f^\prime(x)=4x^{3}+6$.

Answer:

$4x^{3}+6$