Differentiation

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Remember the formula logxy=logy/logx=lny/lnx
∴h(x)=(lnx/ln2)x
lnh(x)=xln(lnx/ln2)
Diff. both sides w.r.t x
[1/h(x)]h'(x)=x(ln2/lnx)(1/x)+ln(lnx/ln2)
[1/h(x)]h'(x)=ln2/lnx+ln(lnx/ln2)
h'(x)=(lnx/ln2)x[ln2/lnx+ln(lnx/ln2)]
h'(x)=(lnx/ln2)x-1+(lnx/ln2)xln(lnx/ln2)

2.x5log2y-10=0
x5(lny/ln2)=10
Diff. both sides w.r.t x
x5[(1/ln2)(1/y)(dy/dx)]+(lny/ln2)(5x4)=0
x[1/yln2](dy/dx)=(lny/ln2) [Divide x4 ]

2008-12-26 12:53:23 補充：
yahoo成日食字=.=
x[1/yln2](dy/dx)=(lny/ln2) [Divide x^4 on both sides]
dy/dx=(lny/ln2)(yln2/x)
=ylny/x

1.Find the derivative of the following function.
h(x) = [log(base2) x ]^x

Sol: h(x) = [log(base2) x ]^x

.......h(x) = x[log(base2)x]

......h'(x) = (x) d/dx [log (base2) x] + [log (base2) x] d/dx (x)

......h'(x) = (x)(1/x) + [log (base2) x] (1)

......h'(x) = 1 + log (base 2) x (The Answer)


2.Find dy/dx.
x^5 log(base2) y - 10 = 0

Sol: ....................................................x^5 log(base2) y - 10 = 0

........(x^5) d/dx [log (base 2) y] + [log (base 2) y]d/dx (x^5) = 0

............................x^5 (1/y) (dy/dx) + [log (base 2) y] (5x^4) = 0

......................................................................(x^5/y) (dy/dx) = -5x^4[ log (base2) y]

.....................dy/dx = [-5x^4y log (base 2) y] / (x^5)

.....................dy/dx = [ -5y log (base 2) y] / x (The Answer)