DIFFERENTIAL CALCULUS Key Answers 1. dy / dx = 2. dy / dx = esinx cosx 3. dy / dx = 0 4. dy / dx = coshx 5. dy / dx = Tanhx 6. f1 ( x ) = ‐ 3 sin x + 4 cosx f1 ( π/2 ) = ‐ 3 + 0 = ‐ 3 8. dy / dx = 7x. 7x6 + x7 7x loge7. 9. dy / dx = πx π x π-1 + xπ. πx logeπ. 10. 1 / x 11. L f1 ( a ) = lt f( a – h ) – f ( a ) hÆ0 ‐h 12. dy / dx = ½ 13. dy / dx = √x
14. dy / dx = e cos x + sinx √
√
18. dy / dx = 2 { 1/ 1 + x² }
19. dy / dx = 1/ 2(1 + x²)
20. dy / dx = 3/ (1 + x²)
23. 1
24. dy/dx = {m cos( m sin-1 √x )/ 4(√sin( m sin-1 √x ) }{ 1/√(1 – x )√x }
27 dy/dx = 1/2.
28. dy/dx = - 1/2.
29. dy / dx = 2/ (1 + x²)
30.dy/dx = - x/y
35.
1/4
36. dy/dx = ( 1 + x )1/x { 1/ x(1+x) + log ( 1 + x ) } + x (1 + x ) { ( 1 + x ) / x + log ( 1 + x ) }
x
37. dy/dx =
{( sin
( sin-1 x )x
-1
x) /x + logx / ( √( 1 - x ² )
{ x/sin
-1
} +
x√( 1 - x ² ) +log(sin -1 x)
}
40. dy/dz = -1
45. dy/dz = 2/3
46. dy/dx = ( log x )x
{ 1/ logx + log(logx) } +
( sin-1 x)sinx
{ (sinx/ sin
-1
x) (√( 1 - x ² ) + cosx log sin-1 x)
}