Question

Encontre as derivadas df/dt da função composta f(x,y) = (-23) + x²y² - 9xy7 com x = 1/t² e y = 1/t³ .


a.
-10t -9 + 207 t -22


b.
-10t -11 + 2187 t2186

c.
-13t -14 + 9(2187 t 2188 )

d.
-10t -11 + 207t -24


e.
-24 t -25 + 9 t -24

Answer 1

Resposta: D) df/dt = - 10t⁻¹¹ + 207t⁻²⁴

 

Vamos lá. Aparentemente a função é:

 

$\sf f(x,y)=-\,23+x^2y^2-9xy^7$

 

Dado que x = 1/t² e y = 1/t³, faça a substituição para colocar f em função de t:

 

$\begin{array}{l}\sf f(t)=-\,23+\bigg(\dfrac{1}{t^2}\bigg)^{\!\!2}\bigg(\dfrac{1}{t^3}\bigg)^{\!\!2}-9\bigg(\dfrac{1}{t^2}\bigg)\bigg(\dfrac{1}{t^3}\bigg)^{\!\!7}\\\\\sf f(t)=-\,23+\bigg(\dfrac{1}{t^4}\bigg)\bigg(\dfrac{1}{t^6}\bigg)-9\bigg(\dfrac{1}{t^2}\bigg)\bigg(\dfrac{1}{t^{21}}\bigg)\\\\\sf f(t)=-\,23+\dfrac{1}{t^{10}}-\dfrac{9}{t^{23}}\\\\\sf f(t)=-\,23+t^{-10}-9t^{-23}\end{array}$

 

Agora calcule df/dt, ou seja, a derivada de f em relação a t. Para facilitar nossas vidas, utilize as seguintes regras:

• $\sf \dfrac{d}{dx}(f\pm g)=\dfrac{df}{dx}\pm\dfrac{dg}{dx}$

• $\sf \dfrac{d}{dx}(a)=0$

• $\sf \dfrac{d}{dx}(ax^n)=n\cdot ax^{n-1}$

.

.

.

 

$\begin{array}{l}\sf\dfrac{df}{dt}=\dfrac{d}{dt}\big(-\,23+t^{-10}-9t^{-23}\big)\\\\\sf\dfrac{df}{dt}=\dfrac{d}{dt}(-\,23)+\dfrac{d}{dt}t^{-10}-\dfrac{d}{dt}9t^{-23}\\\\\sf\dfrac{df}{dt}=0-10\cdot t^{-10-1}+23\cdot9t^{-23-1}\\\\\red{\boldsymbol{\sf\dfrac{df}{dt}=-\,10t^{-11}+207t^{-24}}}\end{array}$

 

Letra D