Let $$ f(x)=\left(x^{3}+2 x+4\right)^{4} $$ $$ f^{\prime}(x)=\square $$ $$ f^{\prime}(2)=\square $$

Question

UpStudy AI · Accepted Answer

To find $$ f'(x) $$ for the function

$$
f(x) = (x^3 + 2x + 4)^4,
$$

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

$$
g'(h(x)) \cdot h'(x).
$$

In this case, let

$$
g(u) = u^4 \quad \text{and} \quad h(x) = x^3 + 2x + 4.
$$

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

1. The derivative of $$ g(u) = u^4 $$ is

$$
g'(u) = 4u^3.
$$

2. The derivative of $$ h(x) = x^3 + 2x + 4 $$ is

$$
h'(x) = 3x^2 + 2.
$$

Now, applying the chain rule:

$$
f'(x) = g'(h(x)) \cdot h'(x) = 4(h(x))^3 \cdot h'(x).
$$

Substituting $$ h(x) $$:

$$
f'(x) = 4(x^3 + 2x + 4)^3 \cdot (3x^2 + 2).
$$

Thus, we have

$$
f'(x) = 4(x^3 + 2x + 4)^3 (3x^2 + 2).
$$

Next, we need to evaluate $$ f'(2) $$:

1. First, calculate $$ h(2) $$:

$$
h(2) = 2^3 + 2 \cdot 2 + 4 = 8 + 4 + 4 = 16.
$$

2. Now calculate $$ h'(2) $$:

$$
h'(2) = 3(2^2) + 2 = 3 \cdot 4 + 2 = 12 + 2 = 14.
$$

3. Now substitute $$ h(2) $$ and $$ h'(2) $$ into $$ f'(x) $$:

$$
f'(2) = 4(h(2))^3 \cdot h'(2) = 4(16)^3 \cdot 14.
$$

Calculating $$ 16^3 $$:

$$
16^3 = 4096.
$$

Now substituting back:

$$
f'(2) = 4 \cdot 4096 \cdot 14 = 16384 \cdot 14.
$$

Calculating $$ 16384 \cdot 14 $$:

$$
16384 \cdot 14 = 229376.
$$

Thus, we have:

$$
f'(x) = 4(x^3 + 2x + 4)^3 (3x^2 + 2)
$$

and

$$
f'(2) = 229376.
$$

Final answers:

$$
f^{\prime}(x) = 4(x^3 + 2x + 4)^3 (3x^2 + 2)
$$

$$
f^{\prime}(2) = 229376.
$$
$$ f^{\prime}(x) = 4(x^3 + 2x + 4)^3 (3x^2 + 2) $$ $$ f^{\prime}(2) = 229376 $$

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