# 12X1 T01 02 differentiating logs

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Published on November 19, 2009

Author: nsimmons

Source: slideshare.net

Differentiating Logarithms

Differentiating Logarithms y  log f  x 

Differentiating Logarithms y  log f  x  dy f  x   dx f  x 

Differentiating Logarithms y  log f  x  y  log a f  x  dy f  x   dx f  x 

Differentiating Logarithms y  log f  x  y  log a f  x  dy f  x  dy f  x    dx f  x  dx log a  f  x 

Differentiating Logarithms y  log f  x  y  log a f  x  dy f  x  dy f  x    dx f  x  dx log a  f  x  e.g. (i) y  log3 x  5

Differentiating Logarithms y  log f  x  y  log a f  x  dy f  x  dy f  x    dx f  x  dx log a  f  x  e.g. (i) y  log3 x  5 dy 3  dx 3 x  5

Differentiating Logarithms y  log f  x  y  log a f  x  dy f  x  dy f  x    dx f  x  dx log a  f  x  e.g. (i) y  log3 x  5 ii  y  log x 3 dy 3  dx 3 x  5

Differentiating Logarithms y  log f  x  y  log a f  x  dy f  x  dy f  x    dx f  x  dx log a  f  x  e.g. (i) y  log3 x  5 ii  y  log x 3 dy 3 dy 3 x 2   3 dx 3 x  5 dx x

Differentiating Logarithms y  log f  x  y  log a f  x  dy f  x  dy f  x    dx f  x  dx log a  f  x  e.g. (i) y  log3 x  5 ii  y  log x 3 dy 3 dy 3 x 2   3 dx 3 x  5 dx x 3  x

Differentiating Logarithms y  log f  x  y  log a f  x  dy f  x  dy f  x    dx f  x  dx log a  f  x  e.g. (i) y  log3 x  5 ii  y  log x 3 OR y  log x 3 2 dy 3 dy 3 x   3 dx 3 x  5 dx x 3  x

Differentiating Logarithms y  log f  x  y  log a f  x  dy f  x  dy f  x    dx f  x  dx log a  f  x  e.g. (i) y  log3 x  5 ii  y  log x 3 OR y  log x 3 2 dy  3 dy 3 x  3 y  3 log x dx 3 x  5 dx x 3  x

Differentiating Logarithms y  log f  x  y  log a f  x  dy f  x  dy f  x    dx f  x  dx log a  f  x  e.g. (i) y  log3 x  5 ii  y  log x 3 OR y  log x 3 2 dy  3 dy 3 x  3 y  3 log x dx 3 x  5 dx x dy 3 3   dx x x

(iii) y  loglog x 

(iii) y  loglog x  1 dy  x dx log x

(iii) y  loglog x  1 dy  x dx log x 1  x log x

(iii) y  loglog x  1 dy  x dx log x 1  x log x iv  y  log x  3 x  2

(iii) y  loglog x  1 dy  x dx log x 1  x log x iv  y  log x  3 x  2 dy  x  31   x  2 1  dx  x  3 x  2

(iii) y  loglog x  1 dy  x dx log x 1  x log x iv  y  log x  3 x  2 dy  x  31   x  2 1  dx  x  3 x  2 2x  5   x  3 x  2

(iii) y  loglog x  1 dy  x dx log x 1  x log x iv  y  log x  3 x  2 OR y  log x  3  log x  2  dy  x  31   x  2 1  dx  x  3 x  2 2x  5   x  3 x  2

(iii) y  loglog x  1 dy  x dx log x 1  x log x iv  y  log x  3 x  2 OR y  log x  3  log x  2  dy  x  31   x  2 1 dy 1 1    dx  x  3 x  2 dx x  3 x  2 2x  5   x  3 x  2

(iii) y  loglog x  1 dy  x dx log x 1  x log x iv  y  log x  3 x  2 OR y  log x  3  log x  2  dy  x  31   x  2 1 dy 1 1    dx  x  3 x  2 dx x  3 x  2 2x  5  x  2   x  3    x  3 x  2  x  3 x  2

(iii) y  loglog x  1 dy  x dx log x 1  x log x iv  y  log x  3 x  2 OR y  log x  3  log x  2  dy  x  31   x  2 1 dy 1 1    dx  x  3 x  2 dx x  3 x  2 2x  5  x  2   x  3    x  3 x  2  x  3 x  2 2x  5   x  3 x  2

v   x  5 y  log    x  2

v  y  log  x  5   x  2  x  21   x  51 dy   x  22 dx x5 x2

v  y  log  x  5   x  2  x  21   x  51 dy   x  22 dx x5 x2 3  x  2    x  22  x  5

v  y  log  x  5   x  2  x  21   x  51 dy   x  22 dx x5 x2 3  x  2    x  22  x  5 3   x  2 x  5

v  y  log  x  5  OR y  log x  5  log x  2   x  2  x  21   x  51 dy   x  22 dx x5 x2 3  x  2    x  22  x  5 3   x  2 x  5

v  y  log  x  5  OR y  log x  5  log x  2   x  2  x  21   x  51 dy  1  1 dy  x  22 dx x  5 x  2  dx x5 x2 3  x  2    x  22  x  5 3   x  2 x  5

v  y  log  x  5  OR y  log x  5  log x  2   x  2  x  21   x  51 dy  1  1 dy  x  22 dx x  5 x  2  dx x5  x  2    x  5  x2  x  5 x  2 3  x  2    x  22  x  5 3   x  2 x  5

v  y  log  x  5  OR y  log x  5  log x  2   x  2  x  21   x  51 dy  1  1 dy  x  22 dx x  5 x  2  dx x5  x  2    x  5  x2  x  5 x  2 3  x  2 3     x  22  x  5  x  2 x  5 3   x  2 x  5

v  y  log  x  5  OR y  log x  5  log x  2   x  2  x  21   x  51 dy  1  1 dy  x  22 dx x  5 x  2  dx x5  x  2    x  5  x2  x  5 x  2 3  x  2 3     x  22  x  5  x  2 x  5 3   x  2 x  5 vi  y  log10 6 x

v  y  log  x  5  OR y  log x  5  log x  2   x  2  x  21   x  51 dy  1  1 dy  x  22 dx x  5 x  2  dx x5  x  2    x  5  x2  x  5 x  2 3  x  2 3     x  22  x  5  x  2 x  5 3   x  2 x  5 vi  y  log10 6 x dy 6  dx log 10 6 x

v  y  log  x  5  OR y  log x  5  log x  2   x  2  x  21   x  51 dy  1  1 dy  x  22 dx x  5 x  2  dx x5  x  2    x  5  x2  x  5 x  2 3  x  2 3     x  22  x  5  x  2 x  5 3   x  2 x  5 vi  y  log10 6 x dy 6  dx log 10 6 x 1  x log 10

v  y  log  x  5  OR y  log x  5  log x  2   x  2  x  21   x  51 dy  1  1 dy  x  22 dx x  5 x  2  dx x5  x  2    x  5  x2  x  5 x  2 3  x  2 3     x  22  x  5  x  2 x  5 3   x  2 x  5 Exercise 12B; 1acf, 2chk, 5acehi, 6b, 7ad, 8acef, 9bd, 10ac, 11, 13a, 14bdfhjl, vi  y  log10 6 x 15b, 18bdf, 19b, 20af*, 21a* dy 6  dx log 10 6 x Exercise 12C; 1bdf, 2, 3, 6, 7a, 8, 11, 1  13, 14, 18* x log 10

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