Let f : R → R be a differentiable function with f 0 = 1 and satisfying the equation f x + y = f x f ' y + f ' x f y for all x ,   y ∈ R . Then, the value of log e ⁡ f 4 is

Question

Let f : R &#8594; R be a differentiable function with f 0 = 1 and satisfying the equation f x + y = f x f ' y + f ' x f y for all x , &#160; y &#8712; R . Then, the value of log e &#8289; f 4 is

Aicharya · Accepted Answer

P x , y : &#160; f x + y = f x f ' y + f ' x f y &#160; &#8704; &#160; x , &#160; y &#8712; R P 0,0 : &#160; f 0 = f 0 f ' 0 + f ' 0 f 0 &#8658; 1 = 2 f ' 0 &#8658; f ' 0 = 1 2 P x , 0 : &#160; f x = f x . f ' 0 + f ' x . f 0 &#8658; &#160; &#160; &#160; f x = 1 2 f x + f ' x &#8658; &#160; &#160; &#160; f ' x = 1 2 f x &#8658; &#160; &#160; &#160; f x = e 1 2 x &#8658; &#160; &#160; &#160; ln &#8289; f 4 = 2

Let f : R → R be a differentiable function with f 0 = 1 and satisfying the equation f x + y = f x f…

Solution

Let f : R &#8594; R be a differentiable function with f 0 = 1 and satisfying the equation f x + y = f x f…

Solution

Let f : R → R be a differentiable function with f 0 = 1 and satisfying the equation f x + y = f x f…