Find the derivative of $$ y=\left(\frac{x^{2}+2}{3}\right)^{3} $$ $$ \frac{d y}{d x}=\square $$

Question

UpStudy AI · Accepted Answer

To find the derivative of the function

$$
y = \left(\frac{x^2 + 2}{3}\right)^3,
$$

we will use the chain rule. The chain rule states that if you have a composite function $$ y = f(g(x)) $$, then the derivative is given by

$$
\frac{dy}{dx} = f'(g(x)) \cdot g'(x).
$$

In this case, let

$$
g(x) = \frac{x^2 + 2}{3}
$$

and

$$
f(u) = u^3,
$$

where $$ u = g(x) $$.

First, we need to find $$ g'(x) $$:

$$
g(x) = \frac{x^2 + 2}{3} \implies g'(x) = \frac{1}{3} \cdot (2x) = \frac{2x}{3}.
$$

Next, we find $$ f'(u) $$:

$$
f(u) = u^3 \implies f'(u) = 3u^2.
$$

Now, we can apply the chain rule:

$$
\frac{dy}{dx} = f'(g(x)) \cdot g'(x) = 3\left(g(x)\right)^2 \cdot g'(x).
$$

Substituting $$ g(x) $$ and $$ g'(x) $$:

$$
\frac{dy}{dx} = 3\left(\frac{x^2 + 2}{3}\right)^2 \cdot \frac{2x}{3}.
$$

Now, simplify this expression:

1. Calculate $$ \left(\frac{x^2 + 2}{3}\right)^2 $$:

$$
\left(\frac{x^2 + 2}{3}\right)^2 = \frac{(x^2 + 2)^2}{9}.
$$

2. Substitute this back into the derivative:

$$
\frac{dy}{dx} = 3 \cdot \frac{(x^2 + 2)^2}{9} \cdot \frac{2x}{3}.
$$

3. Simplify:

$$
\frac{dy}{dx} = \frac{3 \cdot 2x \cdot (x^2 + 2)^2}{27} = \frac{2x (x^2 + 2)^2}{9}.
$$

Thus, the derivative is

$$
\frac{dy}{dx} = \frac{2x (x^2 + 2)^2}{9}.
$$

So, the final answer is

$$
\frac{d y}{d x} = \frac{2x (x^2 + 2)^2}{9}.
$$
$$\frac{d y}{d x} = \frac{2x (x^2 + 2)^2}{9}$$

Find the derivative of