91Ó°ÊÓ

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. $$ f(x, y)=x^{2}+y^{2}, \quad x=t, y=t^{2} $$

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. $$ f(x, y)=\sqrt{x^{2}+y^{2}}, y=t^{2}, x=t $$

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. $$ f(x, y)=x y, x=1-\sqrt{t}, y=1+\sqrt{t} $$

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. $$ f(x, y)=\frac{x}{y}, x=e^{t}, y=2 e^{t} $$

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. $$ f(x, y)=\ln (x+y), \quad x=e^{t}, y=e^{t} $$

For the following exercises, find \(\frac{d f}{d t}\) using the chain rule and direct substitution. $$ f(x, y)=x^{4}, \quad x=t, y=t $$

Let \(w(x, y, z)=x^{2}+y^{2}+z^{2}\) \(x=\cos t, y=\sin t\) and \(z=e^{t}\) Express \(w\) as a function of \(t\) and find \(\frac{d w}{d t}\) directly. Then, find \(\frac{d w}{d t}\) using the chain rule.

Let \(z=x^{2} y, \quad\) where \(x=t^{2}\) and \(y=t^{3} .\) Find \(\frac{d z}{d t}\)

Let \(u=e^{x} \sin y\) where \(x=t^{2}\) and \(y=\pi t\). Find \(\frac{d u}{d t}\) when \(x=\ln 2\) and \(y=\frac{\pi}{4}\)

For the following exerises, find \(\frac{d y}{d x}\) using parial derivatives. $$ \sin (6 x)+\tan (8 y)+5=0 $$