Question

Determine dz/dt for the two following mathematical relations: a) z = xe^xy, x = t^2, y = t^-1 b) z = x^2 y^3 + y cos x, x = ln (t^2), y = sin (4t)

Answer 1

a)
`(dz)/(dt) = e^(t)\cdot (2\cdot t+2\cdot t^(2)-1)`

b)
`(dz)/(dt) = [4\cdot \ln t\cdot \sin^(3) 4t - \sin 4t \cdot \sin (2\ln t)]\cdot \left((2)/(t) \right)+[12\cdot (\ln t)^(2)\cdot \sin^(2) 4t + \cos (2\cdot \ln t)]\cdot (4\cdot \cos 4t)`

Explanation:

In this exercise we use implicit differentiation to define
`(dz)/(dt)`:

a)
`z = x\cdot e^(x\cdot y)`,
`x = t^(2)`,
`y = t^(-1)`

`(dz)/(dt) = (dx)/(dt)\cdot e^(x\cdot y) +x\cdot e^(x\cdot y)\cdot \left[(dx)/(dt)\cdot y+x\cdot (dy)/(dt) \right]`

`(dz)/(dt) = e^(x\cdot y)\cdot \left\{(dx)/(dt)+x\cdot \left[(dx)/(dt)\cdot y+x\cdot (dy)/(dt) \right] \right\}`

`(dz)/(dt) = e^(x\cdot y)\cdot \left[(1+x\cdot y)\cdot (dx)/(dt)+x\cdot (dy)/(dt) \right]`(1)

`(dx)/(dt) = 2\cdot t`,
`(dy)/(dt) = -t^(-2)`

`(dz)/(dt) = e^(t)\cdot \left[(1+t)\cdot 2\cdot t +t^(2)\cdot (-t^(-2))\right]`

`(dz)/(dt) = e^(t)\cdot (2\cdot t+2\cdot t^(2)-1)`

b)
`z = x^(2)\cdot y^(3)+y\cdot \cos x`,
`x = \ln t^(2)`,
`y = \sin (4\cdot t)`

`(dz)/(dt) = 2\cdot x\cdot y^(3)\cdot (dx)/(dt) +x^(2)\cdot (3\cdot y^(2))\cdot (dy)/(dt)+(\cos x)\cdot (dy)/(dt) +y\cdot (-\sin x)\cdot (dx)/(dt)`

`(dz)/(dt) = (2\cdot x\cdot y^(3)-y\cdot \sin x)\cdot (dx)/(dt) + (3\cdot x^(2)\cdot y^(2)+\cos x)\cdot (dy)/(dt)`

`x = \ln t^(2)`

`x = 2\cdot \ln t`

`(dx)/(dt) = (2)/(t)`,
`(dy)/(dt) = 4\cdot \cos 4t`

`(dz)/(dt) = [4\cdot \ln t\cdot \sin^(3) 4t - \sin 4t \cdot \sin (2\ln t)]\cdot \left((2)/(t) \right)+[12\cdot (\ln t)^(2)\cdot \sin^(2) 4t + \cos (2\cdot \ln t)]\cdot (4\cdot \cos 4t)`