Expert: Ahmed Salami - 11/17/2008

Question
y- (((e^sinx)(tan^3)x)/(3x^2-1)^8)
a) use logarithmic differentiation to find y'
b) use quotient and/or product rules to find y'


Answer
Hi Lauren,
This is a bit long to work through and so because of time i would just let you know how to go about it.
y = (e^sinx).[tan^3 (x)]/[(3x^2 - 1)^8]
Finding the natural logarithm of both sides,
ln y = ln {(e^sinx).[tan^3 (x)]/[(3x^2 - 1)^8]}
ln y = ln (e^sinx) + ln [tan^3 (x)] - ln [(3x^2 - 1)^8]
    = sinx + 3ln(tanx) - 8ln(3x^2 - 1)
Note: ln (e^p) = p
Differentiating both sides with respect to x,
1/y(dy/dx) = cosx + (3/tanx).(sec^2 x) - [8/(3x^2 - 1)].6x
dy/dx = y {cosx + (3/tanx).(sec^2 x) - [8/(3x^2 - 1)].6x}

Just take your time to simplify.

Using the product rule, find the derivative of (e^sinx).[tan^3 (x)] and later use the quotient rule to combine with [(3x^2 - 1)^8]. Say, for instance
u = (e^sinx).[tan^3 (x)]
you can use the product rule to find du/dx which is then useful when using the quotient rule with y = u/v

Regards