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MHTCET 11 May 2024 Shift 1 Paper

Question : 110 of 150

Marks: +1, -0

\mathrm{f}(x)=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)

f^{\prime}(1)=

Solution:

Concept:

Use logarithmic differentiation to simplify the derivative of a product.

Explanation:

Let

y=f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)

Take natural logarithm on both sides:

\log y=\log(1+x)+\log(1+x^2)+\log(1+x^4)+\log(1+x^8)

Differentiate both sides with respect to

x

\frac{1}{y}\frac{dy}{dx}=\frac{1}{1+x}+\frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}

x=1

y=f(1)=(2)(2)(2)(2)=16

Substitute

x=1

into the differentiated equation:

\frac{1}{16}f'(1)=\frac{1}{2}+\frac{2}{2}+\frac{4}{2}+\frac{8}{2}=\frac{15}{2}

Therefore:

f'(1)=16 \times \frac{15}{2}=120

Answer:

f'(1)=120

Correct option: D. 120

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