91Ó°ÊÓ

An invertible function \(f(x)\) is given along with a point that lies on its graph. Using Theorem 2.7.1, evaluate \(\left(f^{-1}\right)^{\prime}(x)\) at the indicated value. \(f(x)=x^{3}-6 x^{2}+15 x-2\) Point \(=(1,8)\) Evaluate \(\left(f^{-1}\right)^{\prime}(8)\)

The height \(H\), in feet, of a river is recorded \(t\) hours after midnight, April 1. What are the units of \(H^{\prime}(t) ?\)

Compute the derivative of the given function. $$g(x)=14 x^{3}+7 x^{2}+11 x-29$$

(a) Use the Quotient Rule to differentiate the function. (b) Manipulate the function algebraically and differentiate without the Quotient Rule. (c) Show that the answers from \((\mathrm{a})\) and \((\mathrm{b})\) are equivalent. $$g(x)=\frac{x^{3}-2 x^{2}}{2 x^{2}}$$

Use the definition of the derivative to compute the derivative of the given function. $$f(x)=3 x^{2}-x+4$$

Compute the derivative of the given function. $$m(t)=9 t^{5}-\frac{1}{8} t^{3}+3 t-8$$

Compute the derivative of the given function. $$f(x)=\left(x+\frac{1}{x}\right)^{4}$$

An invertible function \(f(x)\) is given along with a point that lies on its graph. Using Theorem 2.7.1, evaluate \(\left(f^{-1}\right)^{\prime}(x)\) at the indicated value. \(f(x)=\frac{1}{1+x^{2}}, x \geq 0\) Point \(=(1,1 / 2)\) Evaluate \(\left(f^{-1}\right)^{\prime}(1 / 2)\)

(a) Use the Quotient Rule to differentiate the function. (b) Manipulate the function algebraically and differentiate without the Quotient Rule. (c) Show that the answers from \((\mathrm{a})\) and \((\mathrm{b})\) are equivalent. $$h(s)=\frac{3}{4 s^{3}}$$

Use the definition of the derivative to compute the derivative of the given function. $$r(x)=\frac{1}{x}$$