If-z-3-x-2-y-2-dx-dt-3-dy-dt-2-find-dz-dt-when-x-4-and-y-1-

Question Number 125996 by bramlexs22 last updated on 16/Dec/20

I f z^{3} = x^{2} - y^{2}, \to {\begin{cases} \frac{d x}{d t} = 3 \\ \frac{d y}{d t} = 2 \end{cases}

f i n d \frac{d z}{d t} w h e n x = 4 a n d y = 1

Answered by Olaf last updated on 16/Dec/20

z^{3} = x^{2} - y^{2}

\Rightarrow 3 z^{2} \frac{d z}{d t} = 2 x \frac{d x}{d t} - 2 y \frac{d y}{d t} (1)

x = 4, y = 1 \Rightarrow z^{3} = 4^{2} - 1^{2} = 1 \bar{5}, z = 15^{\frac{1}{3}}

(1) : 3 . 15^{\frac{2}{3}} \frac{d z}{d t} = 2 \times 4 \times 3 - 2 \times 1 \times 2 = 20

\frac{d z}{d t} = \frac{20}{3 . 15^{\frac{2}{3}}} = 3^{- \frac{5}{3}} . 4 . 5^{\frac{1}{3}}

Commented by bramlexs22 last updated on 16/Dec/20

t h a n k s