Find the derivative of the following functions by using the appropriate rultof differentiation:
1. y=1/√x[x^2-2/X]
2. h(X)=sin x/1+cos x
3. G(X)=(cos5x)^sin(x^2)
4. F(X) = ∫x√x t√t^2+1dt
1
Expert's answer
2021-08-30T17:27:00-0400
1)y=x(x2−x2)1y=x3−21=(x3−2)2−1differentiate with respect to x, we gety′=2−1(x3−2)2−3.3x2(by chain rule )=2(x3−2)23−3x22)h=1+cosxsinx by usingvuformula and chain rule )h′=(1+cosx)2(1+cosx)cosx−sinx(−sinx)=(1+cosx)2cosx+cos2x+sin2x=(1+cosx)2cosx+1=1+cosx13)g(x)=(cos5x)sin(x2)taking logarithm both side, we get )log(g)=sin(x2)log(cos5x)differentiate with respect to x, we getgg′=sin(x2).(cos5x1).(−sin5x).(5)+log(cos5x).cos(x2).(2x)by chain rule )g′=g[cos5x−5sin(x2)sin5x+2xcos(x2)log(cos5x)]=(cos5x)sin(x2)[cos5x−5sin(x2)sin5x+2xcos(x2)log(cos5x)]4)f=∫xxtt2+1dtby using leibnitz rule for differentiation, we getf′=∫xx∂x∂(tt2+1)dt+dxdx.(xx2+1)−dxdx.(x(x+1))=0+1.(xx2+1)−2x1.(x(x+1))=xx2+1−2x+1
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