Differentiate: $$ f(x)=\left(16 x^{3}-25 x^{6}\right)^{7 / 2} $$

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We define
$$
u(x)=16x^3-25x^6.
$$
Then
$$
f(x)=\left(u(x)\right)^{\frac{7}{2}}.
$$

Using the chain rule, the derivative is
$$
f'(x)=\frac{7}{2}\left(u(x)\right)^{\frac{5}{2}}u'(x).
$$

First, compute $$u'(x)$$:
$$
u'(x)=\frac{d}{dx}\left(16x^3-25x^6\right)=48x^2-150x^5.
$$

Thus, the derivative of $$f(x)$$ is
$$
f'(x)=\frac{7}{2}\left(16x^3-25x^6\right)^{\frac{5}{2}}\left(48x^2-150x^5\right).
$$
The derivative of $$ f(x) $$ is $$ f'(x) = \frac{7}{2} \left(16x^3 - 25x^6\right)^{5/2} \left(48x^2 - 150x^5\right) $$.

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