ln Diff Example - # 2

[Jason76]

\(\displaystyle y = x^{5x}\)

\(\displaystyle \ln[y]= \ln[x^{5x}]\)

\(\displaystyle \ln[y]= \ln[(5x)(x)]\)

\(\displaystyle \ln(y)= \ln(5x) + \ln (x)\)

\(\displaystyle \dfrac{1}{y}y'= \dfrac{5}{x} + \dfrac{1}{x}\)

\(\displaystyle y'= [y]\dfrac{5}{x} + \dfrac{1}{x}\)

\(\displaystyle y'= [ x^{5x}]\dfrac{5}{x} + \dfrac{1}{x}\)

 

[fcabanski]

You're making the same mistake on every one of these problems.

Derivative of \(\displaystyle ln(u^x) = x*ln(u)\) not \(\displaystyle ln(x*u)\)

 

[Jason76]

You're making the same mistake on every one of these problems.

Derivative of \(\displaystyle ln(u^x) = x*ln(u)\) not \(\displaystyle ln(x*u)\)

So you don't use the ln exponent rule ?

 

[Jason76]

\(\displaystyle y = x^{5x}\)

\(\displaystyle \ln[y]= \ln[x^{5x}]\)

\(\displaystyle \ln[y]= [(5x)\ln(x)]\)

\(\displaystyle \dfrac{1}{y} y' = [\ln(x)] [5] + [5x][\dfrac{1}{x}]\)

\(\displaystyle y' = [y] [\ln(x)] [5] + [5x][\dfrac{1}{x}]\)

\(\displaystyle y' = [ x^{5x}] 5\ln(x)] + 5 \)

 

[Deleted member 4993]

\(\displaystyle y = x^{5x}\)

\(\displaystyle \ln[y]= \ln[x^{5x}]\)

\(\displaystyle \ln[y]= \ln[(5x)(x)]\) ..............Incorrect ............should be.......... \(\displaystyle \ln[y]= (5*x)*\ln[(x)]\)

\(\displaystyle \ln(y)= \ln(5x) + \ln (x)\)

\(\displaystyle \dfrac{1}{y}y'= \dfrac{5}{x} + \dfrac{1}{x}\)

\(\displaystyle y'= [y]\dfrac{5}{x} + \dfrac{1}{x}\)

\(\displaystyle y'= [ x^{5x}]\dfrac{5}{x} + \dfrac{1}{x}\)

.

 

[HallsofIvy]

\(\displaystyle y = x^{5x}\)

\(\displaystyle \ln[y]= \ln[x^{5x}]\)

\(\displaystyle \ln[y]= \ln[(5x)(x)]\)

\(\displaystyle \ln(y)= \ln(5x) + \ln (x)\)

\(\displaystyle \dfrac{1}{y}y'= \dfrac{5}{x} + \dfrac{1}{x}\)

\(\displaystyle y'= [y]\dfrac{5}{x} + \dfrac{1}{x}\)

\(\displaystyle y'= [ x^{5x}]\dfrac{5}{x} + \dfrac{1}{x}\)

In addition to the errors others have pointed out, these last two lines should be
\(\displaystyle y'= [y]\left(\dfrac{5}{x}+ \dfrac{1}{x}\right)\)

\(\displaystyle y'= [ x^{5x}]\left(\dfrac{5}{x} + \dfrac{1}{x}\right)\)

 

[Deleted member 4993]

\(\displaystyle y = x^{5x}\)

\(\displaystyle \ln[y]= \ln[x^{5x}]\)

\(\displaystyle \ln[y]= [(5x)\ln(x)]\)

\(\displaystyle \dfrac{1}{y} y' = [\ln(x)] [5] + [5x][\dfrac{1}{x}]\)

\(\displaystyle y' = [y] [\ln(x)] [5] + [5x][\dfrac{1}{x}]\) .............Incorrect ....... should be \(\displaystyle y' = [y]\left( [\ln(x)] [5] + [5x][\dfrac{1}{x}]\right )\)

\(\displaystyle y' = [ x^{5x}] 5\ln(x)] + 5 \)

.

 

[HallsofIvy]

So you don't use the ln exponent rule ?

What do you think the "ln exponent" rule is?