$$ f^{\prime}(x) $$ أوجدي $$ \cdot f(x)=2\left(\frac{2 x^{2}}{3}-3\right)^{6} $$

Question

UpStudy AI · Accepted Answer

To differentiate 
$$
f(x)=2\left(\frac{2x^2}{3}-3\right)^6,
$$
we apply the chain rule.

1. Define
   $$
   g(x)=\frac{2x^2}{3}-3.
   $$
   Then 
   $$
   f(x)=2\,[g(x)]^6.
   $$

2. Differentiate using the chain rule:
   $$
   f'(x)=2\cdot 6\,[g(x)]^{6-1}\cdot g'(x)=12\,[g(x)]^5\cdot g'(x).
   $$

3. Differentiate $$ g(x) $$:
   $$
   g'(x)=\frac{d}{dx}\left(\frac{2x^2}{3}\right)-\frac{d}{dx}(3)=\frac{4x}{3}.
   $$

4. Substitute $$ g(x) $$ and $$ g'(x) $$:
   $$
   f'(x)=12\left(\frac{2x^2}{3}-3\right)^5\cdot \frac{4x}{3}.
   $$

5. Simplify the constants:
   $$
   12\cdot \frac{4x}{3}=\frac{48x}{3}=16x.
   $$

Thus, the derivative is
$$
f'(x)=16x\left(\frac{2x^2}{3}-3\right)^5.
$$
The derivative $$ f'(x) $$ is $$ 16x\left(\frac{2x^2}{3}-3\right)^5 $$.

\( f^{\prime}(x) \) أوجدي \( \cdot f(x)=2\left(\frac{2 x^{2}}{3}-3\right)^{6} \)