Use logarithmic differentiation to find:(adsbygoogle = window.adsbygoogle || []).push({});

a.) d/dx of [(sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2))]

b.) d^2/dx^2 (sech^-1(e^(2*x)))

work shown for a:

let y= [(sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2))]

taking the natural logarithm of both sides:

ln y= ln [(sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2))]

ln y= ln (sin^-1(x^2)*sinh^-1(x^2)) - ln (sin^4(x^2))

ln y = ln (sin^-1(x^2)) +ln (sinh^-1(x^2)) - 4*ln(sin^4(x^2))

differentiating both sides:

1/y* y'= (1/sin^-1(x^2))(1/sqrt(1-(x^2)^2)(2x) + (1/sinh^-1(x^2))(2x)(1/sqrt(1+(x^2)^2)) - (4)(1/sin(x^2))(2x*cos(x^2))

substituting y i got:

y'= (sin^4(x^2)*2x)/(sqrt(1-x^4)) + 2xsin^4(x^2)/(sqrt(1+x^4)) - 8x*cos(x^2)sin^3(x^2)

is this correct... i think I may have made a few errors

work shown for b:

let y= sech^-1(e^(2x))

taking natural logarithm of both sides:

ln y= ln (sech^-1(e^(2x))

differentiating both sides:

y'= 2/(sqrt(1-e^(4x))

to find d^2/dx^2 i set dy/dx as y again:

therefore let y= 2/(sqrt(1-e^(4x))

finding the natural logarithm of both sides of dy/dx

i get ln y= ln 2- (1/2)ln (1-e^(4x))

differentiating both sides leads to:

y"= 4e^(4x)/((1-e^(4x))^(3/2))

is this correct.. i believe i may have made a few errors.. please check

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Logarithmic differentiation help needed work shown

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**