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### Answers (1)

Answer: 120

Hint: To solve this equation we use $\frac{d}{dx}uvwz$   form

Given: \begin{aligned} &f(x)=(1+x) \cdot\left(1+x^{2}\right) \cdot\left(1+x^{4}\right)\left(1+x^{8}\right)\\ \end{aligned}

u                v                   w                z

Solution:

$\frac{d}{d x}(u v w z)=v w z \cdot \frac{d u}{d x}+u w z \cdot \frac{d v}{d x}+u v z \cdot \frac{d w}{d x}+u v w \frac{d z}{d x}$

$\begin{array}{r} f^{\prime}(x)=1 \times\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)+(1+x)\left(1+x^{4}\right)\left(1+x^{8}\right) 2 x+ \\ (1+x)\left(1+x^{2}\right)\left(1+x^{8}\right) 4 x^{3}+8 x^{7}(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right) \end{array}$

\begin{aligned} &f^{\prime}(1)=1 \times 2 \times 2 \times 2+2 \times 2 \times 2 \times 2+2 \times 2 \times 2 \times 4+8 \times 1 \times 2 \times 2 \times 2 \\\\ &=8+16+36+64 \\\\ &f(1)^{\prime}=120 \end{aligned}

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