Question

1. fill in the blanks.

the power chain rule states $\frac{d}{dx}(u^{n})=$ ______________.
the \outside - inside\ rule states $\frac{dy}{dx}=$ ______________, where $y = f(g(x))$.
let $y = f(g(x))$ and $u = g(x)$ so that $y = f(u)$. then the chain rule using leibnizs notation states that $\frac{dy}{dx}=$ ______________.

1. given $y=(3x^{2}-2)^{8}$, find $\frac{dy}{dx}$ by filling in the blanks.

first, let $u = g(x)=$ ____________. this is your ____________ function.
writing $y$ in terms of $u$, we have $y = f(u)=$ ____________. this is your ____________ function.
then, $\frac{du}{dx}=g(x)=$ ____________ and $\frac{dy}{du}=f(u)=$ ____________.
thus, $\frac{dy}{dx}=$ ______________.
the next step is to reverse substitute to obtain a final answer for $\frac{dy}{dx}$ that is in terms of $x$ (no $u$ left in the final answer.) $\frac{dy}{dx}=$ ____________ = ____________.

Explanation:

Step1: Recall Power - Chain Rule

The Power Chain Rule states $\frac{d}{dx}(u^{n})=nu^{n - 1}\frac{du}{dx}$.

Step2: Recall Outside - Inside Rule

The "Outside - Inside" Rule states $\frac{dy}{dx}=f'(g(x))g'(x)$ where $y = f(g(x))$.

Step3: Recall Chain Rule in Leibniz's notation

If $y = f(g(x))$ and $u = g(x)$ so that $y = f(u)$, then the Chain Rule using Leibniz's notation states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

Step4: For $y=(3x^{2}-2)^{8}$

Let $u = g(x)=3x^{2}-2$. This is the inside function.

Step5: Rewrite $y$ in terms of $u$

Writing $y$ in terms of $u$, we have $y = f(u)=u^{8}$. This is the outside function.

Step6: Differentiate $u$ and $y$ with respect to appropriate variables

$\frac{du}{dx}=g'(x)=6x$ and $\frac{dy}{du}=f'(u)=8u^{7}$.

Step7: Apply Chain Rule

Thus, $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=8u^{7}\cdot6x$.

Step8: Reverse - substitute $u$

The final answer for $\frac{dy}{dx}$ in terms of $x$ is $8(3x^{2}-2)^{7}\cdot6x = 48x(3x^{2}-2)^{7}$.

Answer:

1. $\frac{d}{dx}(u^{n})=nu^{n - 1}\frac{du}{dx}$
2. $\frac{dy}{dx}=f'(g(x))g'(x)$
3. $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$
4. $u = g(x)=3x^{2}-2$, inside
5. $y = f(u)=u^{8}$, outside
6. $\frac{du}{dx}=g'(x)=6x$, $\frac{dy}{du}=f'(u)=8u^{7}$
7. $\frac{dy}{dx}=8u^{7}\cdot6x$
8. $\frac{dy}{dx}=48x(3x^{2}-2)^{7}$