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Method of Differentiation: Suppose Function Derivative Derivative Derivative Function 1

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

Question
Suppose the function f(x)−f(2x)f(x) - f(2x) has the derivative 55 at x=1x = 1 and derivative 77 at x=2x = 2. The derivative of the function f(x)−f(4x)−10xf(x) - f(4x) - 10x at x=1x = 1 is equal to.
Solution
Answer: 9
Step 1: Set up equations from the given information Let g(x)=f(x)−f(2x)g(x) = f(x) - f(2x). Then g′(x)=f′(x)−2f′(2x)g'(x) = f'(x) - 2f'(2x).
g′(1)=f′(1)−2f′(2)=5⋯(1)g'(1) = f'(1) - 2f'(2) = 5 \quad \cdots (1)
g′(2)=f′(2)−2f′(4)=7⋯(2)g'(2) = f'(2) - 2f'(4) = 7 \quad \cdots (2)
 Step 2: Compute the required derivative Let h(x)=f(x)−f(4x)−10xh(x) = f(x) - f(4x) - 10x. Then h′(1)=f′(1)−4f′(4)−10h'(1) = f'(1) - 4f'(4) - 10. From (2)(2): f′(4)=f′(2)−72f'(4) = \dfrac{f'(2)-7}{2}. Substituting:
f′(1)−4⋅f′(2)−72−10=f′(1)−2f′(2)+14−10=5+4=9f'(1) - 4\cdot\frac{f'(2)-7}{2} - 10 = f'(1) - 2f'(2) + 14 - 10 = 5 + 4 = 9
 Answer: 9
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