84. Let $$ r(x)=f(g(h(x))) $$, where $$ h(1)=2, g(2)=3, h^{\prime}(1)=4 $$, $$ g^{\prime}(2)=5 $$, and $$ f^{\prime}(3)=6 $$. Find $$ r^{\prime}(1) $$. 85. Suppose $$ g $$ is a twice differentiable function (the first derivative $$ g^{\prime} $$ and the second derivative $$ g^{\prime \prime} $$ exist) and $$ f(x)=x g\left(x^{2}\right) $$. Find $$ f^{\prime \prime} $$ in terms of $$ g, g^{\prime} $$, and $$ g^{\prime \prime} $$. 86. Suppose $$ F(x)=f(3 f(4 f(x))) $$, where $$ f(0)=0 $$ and $$ f^{\prime}(0)=2 $$. Find $$ F^{\prime}(0) $$.

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Claro, aquí tienes las soluciones a los problemas 84, 85 y 86 en español:

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**Problema 84:**

Sea $$ r(x) = f(g(h(x))) $$. Queremos encontrar la derivada $$ r'(1) $$.

Usando la regla de la cadena:

$$
r'(x) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)
$$

Evaluamos en $$ x = 1 $$:

$$
h(1) = 2
$$
$$
g(h(1)) = g(2) = 3
$$
$$
f'(g(h(1))) = f'(3) = 6
$$
$$
g'(h(1)) = g'(2) = 5
$$
$$
h'(1) = 4
$$

Entonces:

$$
r'(1) = f'(3) \cdot g'(2) \cdot h'(1) = 6 \cdot 5 \cdot 4 = 120
$$

**Respuesta:** $$ r'(1) = 120 $$

---

**Problema 85:**

Dada la función $$ f(x) = x \cdot g(x^2) $$, donde $$ g $$ es dos veces diferenciable, queremos encontrar $$ f''(x) $$ en términos de $$ g $$, $$ g' $$ y $$ g'' $$.

Primero, calculamos la primera derivada usando la regla del producto y la regla de la cadena:

$$
f'(x) = g(x^2) + x \cdot g'(x^2) \cdot 2x = g(x^2) + 2x^2 g'(x^2)
$$

Ahora, calculamos la segunda derivada:

$$
f''(x) = \frac{d}{dx} \left[ g(x^2) + 2x^2 g'(x^2) \right]
$$

Derivando término a término:

$$
\frac{d}{dx} g(x^2) = 2x \cdot g'(x^2)
$$
$$
\frac{d}{dx} \left( 2x^2 g'(x^2) \right) = 4x \cdot g'(x^2) + 4x^3 \cdot g''(x^2)
$$

Sumando los términos derivados:

$$
f''(x) = 2x g'(x^2) + 4x g'(x^2) + 4x^3 g''(x^2) = 6x g'(x^2) + 4x^3 g''(x^2)
$$

**Respuesta:**

$$
f''(x) = 6x \cdot g'(x^2) + 4x^3 \cdot g''(x^2)
$$

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**Problema 86:**

Dada la función $$ F(x) = f(3 f(4 f(x))) $$, donde $$ f(0) = 0 $$ y $$ f'(0) = 2 $$, queremos encontrar $$ F'(0) $$.

Aplicamos la regla de la cadena múltiples veces:

$$
F'(x) = f'\left( 3 f(4 f(x)) \right) \cdot 3 f'\left( 4 f(x) \right) \cdot 4 f'(x)
$$

Evaluamos en $$ x = 0 $$:

$$
f(0) = 0
$$
$$
f(4 f(0)) = f(0) = 0
$$
$$
f(3 f(4 f(0))) = f(0) = 0
$$
$$
f'\left( 3 f(4 f(0)) \right) = f'(0) = 2
$$
$$
f'\left( 4 f(0) \right) = f'(0) = 2
$$
$$
f'(0) = 2
$$

Entonces:

$$
F'(0) = 2 \cdot 3 \cdot 2 \cdot 4 \cdot 2 = 96
$$

**Respuesta:** $$ F'(0) = 96 $$

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**Problema 84:** $$ r'(1) = 120 $$ **Problema 85:** $$ f''(x) = 6x \cdot g'(x^2) + 4x^3 \cdot g''(x^2) $$ **Problema 86:** $$ F'(0) = 96 $$

Let , where ,
, and . Find .

Suppose is a twice differentiable function (the first
derivative and the second derivative exist) and
. Find in terms of , and .

Suppose , where and .
Find .

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