Method of DifferentiationmediumFree

Method of Differentiation: Question

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

Question
If F(x)=f(x)ϕ(x)F(x) = f(x)\phi(x) and f′(x)ϕ′(x)=cf'(x)\phi'(x) = c, then f′′(x)f(x)+ϕ′′(x)ϕ(x)+2cf(x)ϕ(x)\dfrac{f''(x)}{f(x)} + \dfrac{\phi''(x)}{\phi(x)} + \dfrac{2c}{f(x)\phi(x)} is:
AF′′(x)F(x)\dfrac{F''(x)}{F(x)}correct
B2F′′(x)F(x)2\dfrac{F''(x)}{F(x)}
C−F′′(x)F(x)-\dfrac{F''(x)}{F(x)}
D−2F′′(x)F(x)-2\dfrac{F''(x)}{F(x)}
Solution
Step 1: First and second derivatives of the product
F′(x)=f(x)ϕ′(x)+f′(x)ϕ(x)F'(x) = f(x)\phi'(x) + f'(x)\phi(x)
F′′(x)=f(x)ϕ′′(x)+2f′(x)ϕ′(x)+f′′(x)ϕ(x)F''(x) = f(x)\phi''(x) + 2f'(x)\phi'(x) + f''(x)\phi(x)
 Step 2: Divide by F(x)=f(x)ϕ(x)F(x) = f(x)\phi(x)
F′′(x)F(x)=f′′(x)f(x)+ϕ′′(x)ϕ(x)+2f′(x)ϕ′(x)f(x)ϕ(x)\frac{F''(x)}{F(x)} = \frac{f''(x)}{f(x)} + \frac{\phi''(x)}{\phi(x)} + \frac{2f'(x)\phi'(x)}{f(x)\phi(x)}
 Step 3: Apply f′(x)ϕ′(x)=cf'(x)\phi'(x) = c
F′′(x)F(x)=f′′(x)f(x)+ϕ′′(x)ϕ(x)+2cf(x)ϕ(x)\frac{F''(x)}{F(x)} = \frac{f''(x)}{f(x)} + \frac{\phi''(x)}{\phi(x)} + \frac{2c}{f(x)\phi(x)}
 Answer: (1)
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