數學問題 differentiation

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a)find dy/dx

1) y=2x-cos2x

Sol

dy/dx=d2x/dx－dcos2x/dx

=2－(dcos2x/2x)*(d2x/dx)

=2+2sin2x



2) y=(x^2+x)(3x+2)

Sol

dy/dx=d(x^2+x)(3x+2)/dx

=d(3x^3+5x^2+2x)/dx

=9x^2+10x+3



3) y=(1+6x)^1/2

Sol

dy/dx=d(1+6x)^(1/2)/dx

=[d(1+6x)^(1/2)/d(1+6x)][d(1+6x)/dx]]

=3(1+6x)^(－1/2)



b)differentiate the following wiht respect to x.

1) (x－4)^1/2(x^2－2)^5

Sol

d(x－4)^(1/2)*(x^2－2)^5/dx

=(x－4)^(1/2)*[d(x^2－2)^5/dx]+(x^2－2)^5*[d(x－4)^(1/2)/dx]

=(x-4)^(1/2)*5(x^2－2)^4(2x)+(x^2－2)^5*(1/2)(x－4)^(－1/2)


=(x-4)^(－1/2)(x^2－2)^4[10x(x-4)+(1/2)(x^2－2)]

=(x^2－2)^4[20x^2－80x+x^2－2]/[2(x－4)^(1/2)]

=(x^2－2)^4(21x^2－80x－2)/[2(x－4)^(1/2)]



2) (x^2+6)/(x^3+x－1)^1/2

Sol

d(x^2+6)/(x^2+x－1)^(1/2)/dx

=d(x^2-6)(x^2+x－1)^(－1/2)/dx

=(x^2-6)[d(x^2+x－1)^(－1/2)/d(x^2+x－1)]*[d(x^2+x－1)/dx]

+(x^2+x－1)^(－1/2)*[d(x^2－6)/dx]

=(x^2-6)(-1/2)(x^2+x－1)^(－3/2)*(2x+1)+(x^2+x－1)^(－1/2)*(2x)

=(x^2+x－1)^(－3/2)/2[－(x^2－6)(2x+1)+(x^2+x－1)(4x)]

=(x^2+x－1)^(－3/2)/2[－2x^3－x^2+12x+6+4x^3+4x^2－4x]

=(2x^3+3x^2+8x+6)/[2(x^2+x－1)^(3/2)]