4. ( 6 pts) Find the derivative of each of the following functions. You need not take the time to simplify your derivatives. a. $$ f(x)=4\left(3 x^{2}+2^{-6 x}\right)^{10} $$

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Let 
$$
u(x)=3x^2+2^{-6x},
$$
so that 
$$
f(x)=4\left(u(x)\right)^{10}.
$$

Using the chain rule, we have
$$
f'(x)=4\cdot10\left(u(x)\right)^9\cdot u'(x)=40\left(3x^2+2^{-6x}\right)^9u'(x).
$$

Now, differentiate $$u(x)$$ term by term.

1. For $$3x^2$$:
$$
\frac{d}{dx}\left[3x^2\right]=6x.
$$

2. For $$2^{-6x}$$:  
Recall that if $$g(x)=a^{h(x)}$$ then
$$
g'(x)=a^{h(x)}\ln(a)h'(x).
$$
Here, $$a=2$$ and $$h(x)=-6x$$. Therefore,
$$
h'(x)=-6,
$$
and 
$$
\frac{d}{dx}\left[2^{-6x}\right]=2^{-6x}\ln(2)(-6)=-6\ln(2)2^{-6x}.
$$

Combining these, we get
$$
u'(x)=6x-6\ln(2)2^{-6x}.
$$

Substitute $$u'(x)$$ into the expression for $$f'(x)$$:
$$
f'(x)=40\left(3x^2+2^{-6x}\right)^9\left(6x-6\ln(2)2^{-6x}\right).
$$
The derivative of $$ f(x) = 4\left(3x^2 + 2^{-6x}\right)^{10} $$ is: $$ f'(x) = 40\left(3x^2 + 2^{-6x}\right)^9 \left(6x - 6\ln(2)2^{-6x}\right). $$

( 6 pts) Find the derivative of each of the following functions.
You need not take the time to simplify your derivatives.
a.

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