Let `f,g, h` be differentiable functions of x. if `Delta = |(f,g,h),((xf)',(xg)',(xh)'),((x^(2) f)'',(x^(2)g)'',(x^(2) h)'')|and, Delta' = |(f,g,h),(f',g',h'),((x^(n)f'')',(x^(n) g'')',(x^(n) h'')')|`, then n =

If f,g and h are differentiable functions of x and Delta = |{:(f , g , h), ((xf)' , (xg)' , (xh)'), ((x^(2) f)'' , (x^(2) g)'', (x^(2) h)''):}| then prove that Delta'=|{:(f , g , h), (f' , g' , h'), ((x^(3) f'')' , (x^(3) g'')', (x^(3) h)'')':}| .

Let f be a twice differentiable function such that f''(x)=-f(x),a n df^(prime)(x)=g(x),h(x)=[f(x)]^2+[g(x)]^2dot Find h(10)ifh(5)=11

Let f(x)=x^(2) and g(x)=2x+1 be two real functions. Find (f+g) (x), (f-g) (x), (fg) (x), (f/g) (x) .

Let g be the inverse function of f and f'(x)=(x^(10))/(1+x^(2)). If g(2)=a then g'(2) is equal to

Let f:R to R and h:R to R be differentiable functions such that f(x)=x^(3)+3x+2,g(f(x))=x and h(g(x))=x for all x in R . Then, h'(1) equals.

If y= |(f(x),g(x),h(x)),(l,m,n),(a,b,c)| , prove that (dy)/(dx)= |(f'(x),g'(x),h'(x)),(l,m,n),(a,b,c)|

Given (f(x) =log_(10) x and g(x) = e^(piix) . phi (x) =|{:(f(x).g(x),(f(x))^(g(x)),1),(f(x^(2)).g(x^(2)),(f(x^(2)))^(g(x^(2))),0),(f(x^(3)).g(x^(3)),(f(x^(3)))^(g(x^(3))),1):}| the value of phi (10), is

If f'(x)=g'(x) , then . . . . . . .

If f(x)=(a x^2+b)^3, then find the function g such that f(g(x))=g(f(x))dot

If f and g are two real valued functions defined as f(x)= 2x+1 and g(x)= x^(2)+1 , then find f+g