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 \(\frac{d x}{d t}\). $$x^{2} y^{2}=2 y^{3}+1$$

Find the point(s) on the graph of \(y=\left(2 x^{4}+1\right)(x-5)\) where the slope is 1 .

Compute \(\frac{d}{d x} f(g(x))\), where \(f(x)\) and \(g(x)\) are the following: $$f(x)=x(x-2)^{4}, g(x)=x^{3}$$

A point is moving along the graph of \(x^{2}-4 y^{2}=9 .\) When the point is at \((5,-2)\), its \(x\) -coordinate is increasing at the rate of 3 units per second. How fast is the \(y\) -coordinate changing at that moment?

Find \(\frac{d^{2} y}{d x^{2}}\). $$y=\left(x^{2}+1\right)^{4}$$

Compute \(\frac{d y}{d x}\) using the chain rule in formula (1). $$y=u^{3 / 2}, u=4 x+1$$

Compute \(\frac{d y}{d x}\) using the chain rule in formula (1). $$y=\sqrt{u+1}, u=2 x^{2}$$

Find \(\frac{d^{2} y}{d x^{2}}\). $$y=\sqrt{x^{2}+1}$$

A point is moving along the graph of \(x^{3} y^{2}=200 .\) When the point is at \((2,5)\), its \(x\) -coordinate is changing at the rate of \(-4\) units per minute. How fast is the \(y\) -coordinate changing at that moment?

Suppose that the price \(p\) (in dollars) and the weekly sales \(x\) (in thousands of units) of a certain commodity satisfy the demand equation $$ 2 p^{3}+x^{2}=4500 $$ Determine the rate at which sales are changing at a time when \(x=50, p=10\), and the price is falling at the rate of $$\$ .50$$ per week.