Method of DifferentiationmediumFree

Method of Differentiation — JEE Maths practice question

JEE Maths question with a full step-by-step solution.

Question

Let

f_{n+1}(x) = e^{f_n(x)}

for all

1 \le n \le 9

and

f_1(x) = e^x

. Then the first derivative of

f_n(x)

for

n = 10

is:

f_9(x)\, f_{10}(x)

\prod_{r=1}^{9} f_r(x)\, e^x

\prod_{r=1}^{10} e^x f_r(x)

\prod_{r=1}^{10} f_r(x)

correct

Solution

Step 1: Differentiate the recurrence

f'_{n+1}(x) = e^{f_n(x)} f'_n(x) = f_{n+1}(x)\, f'_n(x)

Step 2: Iterate the relation

f'_{10}(x) = f_{10}(x)\, f'_9(x) = f_{10}(x) f_9(x)\, f'_8(x) = \cdots = f_{10}(x) f_9(x) \cdots f_2(x)\, f'_1(x)

Step 3: Use

f'_1(x) = e^x = f_1(x)

f'_{10}(x) = f_{10}(x) f_9(x) \cdots f_2(x) f_1(x) = \prod_{r=1}^{10} f_r(x)

Answer: (4)

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