Using the Chain rule, $$ \frac{d}{d x}\left(\left(2 x^{3}+7 x+1\right)^{-4}\right)=-4\left(2 x^{3}+7 x+1\right)^{-5}\left(6 x^{2}+7\right) $$ Is the derivative TRUE or FALSE? Select one: True False

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The function is given by
$$
f(x) = \left(2x^3 + 7x + 1\right)^{-4}.
$$
We can apply the chain rule by letting
$$
u(x) = 2x^3 + 7x + 1 \quad \text{and} \quad f(u) = u^{-4}.
$$
The derivative of $$ f(u) $$ with respect to $$ u $$ is
$$
f'(u) = -4u^{-5}.
$$
The derivative of $$ u(x) $$ with respect to $$ x $$ is
$$
u'(x) = \frac{d}{dx} \left(2x^3 + 7x + 1\right) = 6x^2 + 7.
$$
According to the chain rule, the derivative of $$ f(x) $$ with respect to $$ x $$ is
$$
f'(x) = f'(u(x)) \cdot u'(x) = -4\left(2x^3 + 7x + 1\right)^{-5} \cdot (6x^2 + 7).
$$
This exactly matches the derivative given in the problem statement.

True

The derivative is True.

Using the Chain rule,

Is the derivative TRUE or FALSE?
Select one:
True
False

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Using the Chain rule, Is the derivative TRUE or FALSE? Select one: True False