If f(x) + x2[f(x)]3 = 10 and f(1) = 2. find f ‘(1).

Answer

Here you the value of f(1) , so to obtain f ‘(1)
……differentiate the given equation on both side with respect to
x ….

Let ….

f(x)+ x^2[ f(x)^3]=10 ……………….. (1) be the equation
…

therefore applying product rule of differentiation …..i.e ….
d(f(x).g(x)) = f(x).d(g(x)) + g(x).d(f(x))

we get ,

f ‘(x) + 2x[f(x)^3] + [x^2.3.f(x)^2.f ‘(x)] = 0

now put x=1 ,we get

f ‘(1) + 2(1)[ f(1)^3] + [ (1)^2.3.f(1)^2.f ‘(1)]=0

and we know f(1)=2

therefore …

f ‘(1) + 2(2^3) + [1.3.(2^2).f ‘(1)] = 0

=> f ‘(1) + 2(8) + (3.4)f ‘(1) = 0

=> f ‘(1) + 16 + 12.f ‘(1) = 0

=> 13.f ‘(1) = -16

=> f ‘(1)= -16/13

and that’s your answer !

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