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- Mathematics 133
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- Ch. 6 Derivative of log and natural log

Anonymous

8/26 Ch6. :Properties and Derivatives of Logarithms Properties of log: Product Rule: logb(xy) = logbx + logby. Quotient rule: logb(x/y) = logbx - logby. Power rule: logb(xn) = n logbx. Definition: logbx = logax / logab. Reciprocal Rule: logb1 = 0. logbb = 1. Logex=lnx Log10x=logx Derivatives: I=open interval of R, I= (a, b) F: R x if I f(x)=lim h0 (f(x+h)-f(x))/h Rules: (cf)’(x)=cf‘(x), c=constant (f+g)=f’(x)+g’(x) (f/g)’(x) =( f’(x)+g’(x))/(g(x))2 Ex. (xn)=nxn-1 (1/x)’=(f◦g)’(x) = f’(y(x))y ‘(x) = -1/y2(x)y’(x) Ex. f(x)=x2sec(1/x) = x2/cos(1/x) F’(x) = 2xsec(1/x) +x2(1/cos(1/x)’ F’(x)=2xsec(1/x) +x2 – (cos(1/x)’/cos2(1/x) F’(x)=2xsec(1/x) -x2 ((-sin(1/x)(-1/x2))/cos21/x F’(x)=2xsec(1/x)-sin(1/x)/cos2(1/x) F’(x)=2xsec(1/x)-tan(1/x)sec(1/x) (ex)’=ex (lnx)’=1/x (ax)’ = (lnx/lna)’ = 1/xlna Ex. F(x)=5cos(x) F(x)’=(ln5)*5cos(x)(cosx)’=-ln5*5cosxsinx G(x)=xsinx G’(x)=(xsinx)’=(esinxlnx)’ =esinxlnx(sinx)’(lnx )+(sinx(lnx)’ G’(x)=xsinx(cosxlnx+sinx/x)

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