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Need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 51

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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


          \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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