91Ó°ÊÓ

Given the function \(f,\) find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of \(f^{-1}\) $$f(x)=-x^{2}+8 ;(7,1)$$

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point. $$x^{2 / 3}+y^{2 / 3}=2 ;(1,1)$$

Calculate the derivative of the following functions. $$y=\sqrt{(3 x-4)^{2}+3 x}$$

Evaluate the derivative of the following functions at the given point. $$A=\pi r^{2} ; r=3$$

Find \(y^{\prime \prime}\) for the following functions. $$y=\tan x$$

Find \(f^{\prime}(x), f^{\prime \prime}(x),\) and \(f^{\prime \prime \prime}(x)\) for the following functions. $$f(x)=\frac{x^{2}-7 x-8}{x+1}$$

Implicit differentiation with rational exponents Determine the slope of the following curves at the given point. $$x \sqrt[3]{y}+y=10 ;(1,8)$$

Find \(\left(f^{-1}\right)^{\prime}(3)\) if \(f(x)=x^{3}+x+1\)

A store manager estimates that the demand for an energy drink decreases with increasing price according to the function \(d(p)=\frac{100}{p^{2}+1},\) which means that at price \(p\) (in dollars), \(d(p)\) units can be sold. The revenue generated at price \(p\) is \(R(p)=p \cdot d(p)\) (price multiplied by number of units). a. Find and graph the revenue function. b. Find and graph the marginal revenue \(R^{\prime}(p)\). c. From the graphs of \(R\) and \(R^{\prime}\), estimate the price that should be charged to maximize the revenue.

Find \(y^{\prime \prime}\) for the following functions. $$y=\sec x \csc x$$