Most Important Selected Qs for JEE AdvancedMathematicsDifferentiation

If x ₀^x (f(t)) d t=(x+2) ₀^x t (f(t)) d t where x>0 , then find the value of f^ (x) (f(x))+ 3 1+x

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Step-by-step solution

Differentiate both sides aligned & x (f(x))+ ₀^x (f(t)) d t=(x+2) (f(x))+ ₀^x t (f(t)) d t & x^2 (f(x))+x (f(x))= ₀^x (f(t)) d t- ₀^x t (f(t)) d t & aligned Again differentiate aligned & x^2 (f(x)) f^ (x)+2 x (f(x))+ (f(x))+x (f(x)) f^ (x)= (f(x))-x (f(x)) & =x(x+1) (f(x))+3 x (f(x))=0 & f^ (x) (f(x))+ 3 1+x =0 aligned

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