91Ó°ÊÓ

Suppose that \(x\) and \(y\) are both differentiable functions of \(t\) and are related by the given equation. Use implicit differentiation with respect to \(t\) to determine \(\frac{d y}{d t}\) in terms of \(x, y\), and \(\frac{d x}{d t}\). $$y^{4}-x^{2}=1$$

Compute \(\frac{d}{d x} f(g(x))\), where \(f(x)\) and \(g(x)\) are the following: $$f(x)=\frac{1}{1+\sqrt{x}}, g(x)=\frac{1}{x}$$

Compute \(\frac{d}{d x} f(g(x))\), where \(f(x)\) and \(g(x)\) are the following: $$f(x)=x^{4}-x^{2}, g(x)=x^{2}-4$$

Find all \(x\) such that \(\frac{d y}{d x}=0\), where $$ y=\left(x^{2}-4\right)^{3}\left(2 x^{2}+5\right)^{5} . $$

Suppose that \(x\) and \(y\) are both differentiable functions of \(t\) and are related by the given equation. Use implicit differentiation with respect to \(t\) to determine \(\frac{d y}{d t}\) in terms of \(x, y\), and \(\frac{d x}{d t}\). $$3 x y-3 x^{2}=4$$

Compute \(\frac{d}{d x} f(g(x))\), where \(f(x)\) and \(g(x)\) are the following: $$f(x)=\frac{4}{x}+x^{2}, g(x)=1-x^{4}$$

Suppose that \(x\) and \(y\) are both differentiable functions of \(t\) and are related by the given equation. Use implicit differentiation with respect to \(t\) to determine \(\frac{d y}{d t}\) in terms of \(x, y\), and \(\frac{d x}{d t}\). $$y^{2}=8+x y$$

Suppose that \(x\) and \(y\) are both differentiable functions of \(t\) and are related by the given equation. Use implicit differentiation with respect to \(t\) to determine \(\frac{d y}{d t}\) in terms of \(x, y\), and \(\frac{d x}{d t}\). $$x^{2}+2 x y=y^{3}$$

Find the point(s) on the graph of \(y=\left(x^{2}+3 x-1\right) / x\) where the slope is \(\underline{5}\).

Compute \(\frac{d}{d x} f(g(x))\), where \(f(x)\) and \(g(x)\) are the following: $$f(x)=\left(x^{3}+1\right)^{2}, g(x)=x^{2}+5$$