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Logarithmic differentiation

Tags
Cegep1
Mathematics
Word count
371 words
Reading time
3 minutes

Used to simplify differentiation or differentiate otherwise impossible equations

  1. Preferably convert explicit differentiation to implicit by replacing f(x) by y
  2. Apply ln
  3. Simplify
  4. Differentiate
  5. Solve for the derivative

Examples

Find y when y=xx.

y=xxlny=lnx=xlnxddxlny=ddx(xlnx)yy=lnx+x1x=lnx+1y=y(lnx+1)=xx(lnx+1)

Find y when y=xxx.

y=xxxlny=lnxxxlny=xxlnxlnlny=ln(xxlnx)=lnxx+lnlnx=xlnx+lnlnxddxlnlny=ddx(xlnx+lnlnx)(lny)lny=lnx+x1x+(lnx)lnxyylny=lnx+1+xxlnxy=ylny(lnx+1+1xlnx)=xxxlnxxx(lnx+1+1xlnx)

Find the derivative of f(x)=xex2(x7+2)3cos2x3.

y=xex2(x7+2)3cos2x3lny=lnxex2(x7+2)3cos2x3=13(ln(xex2(x7+2)3)lncos2x)=13(lnx+lnex2+ln(x7+2)3lncos2x)=13(lnx+x23ln(x7+2)lncos2x)ddxlny=ddx(13(lnx+x23ln(x7+2)lncos2x))yy=13(xx+2x3(x7+2)x7+2(cos2x)cos2x)=13(12x1/2x+2x21x6x7+2+2cosxsinxcos2x)y=xex2(x7+2)3cos2x313(12x1/2x+2x21x6x7+2+2cosxsinxcos2x)

Contributors

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