Question

find du/dt if u=log(x+y+z) where x= cost ,y= sin^2 t and z= cos^2 t​

Answer 1

Answer:

find du/dt if u=log(x+y+z) where x= cost ,y= sin^2 t and z= cos^2 t

Answer 2

Answer:

The result is:

$\dfrac{-sint}{cost+1}$

Step-by-step explanation:

Given:

$u=log(x+y+z)$

where $x=cost$,$y=sin^2t$,$z=cos^2t$

Hence differentiating by substituting the given values:

$u=log(cost+sin^2t+cos^2t)$

Let:

$cost+sin^2t+cos^2t=m$

Applying limit chain rule:

$\dfrac{du}{dt}=\dfrac{du}{dm}\times \dfrac{dm}{dt}$

Hence:

$\dfrac{du}{dt}=\dfrac{d}{dm}[log(m)]\times \dfrac{d}{dt}[cost+sin^2t+cos^2t]$

                           $=\dfrac{1}{m}\times [-sint+2sint\cdot cost+2cost(-sint)]$

Substituting the value of m.

                           $=\dfrac{-sint}{cost+sin^2t+cos^2t}$

As we know the formula $sin^2x+cos^2x=1$

Plugging this in above equation we get the result:

$=\dfrac{-sint}{cost+1}$