Question

find the derivatives of the following functions. g(x) = 2x^3 - 7x^2 + 9 f(x) = (2x^3 - 7x^2 + 9)^6 g(x) = f(x) =

Explanation:

Step1: Apply power - rule to $g(x)$

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $g(x)=2x^{3}-7x^{2}+9$, we have:
$g^\prime(x)=\frac{d}{dx}(2x^{3})-\frac{d}{dx}(7x^{2})+\frac{d}{dx}(9)$.
$\frac{d}{dx}(2x^{3}) = 2\times3x^{2}=6x^{2}$, $\frac{d}{dx}(7x^{2})=7\times2x = 14x$, and $\frac{d}{dx}(9)=0$. So $g^\prime(x)=6x^{2}-14x$.

Step2: Apply chain - rule to $f(x)$

The chain - rule states that if $y = u^n$ and $u = h(x)$, then $\frac{dy}{dx}=n\cdot u^{n - 1}\cdot h^\prime(x)$. Let $u = 2x^{3}-7x^{2}+9$ and $n = 6$. We know from Step 1 that $u^\prime=g^\prime(x)=6x^{2}-14x$. Then $f^\prime(x)=6(2x^{3}-7x^{2}+9)^{5}\cdot(6x^{2}-14x)$.

Answer:

$g^\prime(x)=6x^{2}-14x$
$f^\prime(x)=6(6x^{2}-14x)(2x^{3}-7x^{2}+9)^{5}$