Problem 10
(a) Use the Product Rule to differentiate the function. (b) Manipulate the function algebraically and differentiate without the Product Rule. (c) Show that the answers from (a) and (b) are equivalent. $$f(x)=\left(x^{2}+5\right)\left(3-x^{3}\right)$$
Problem 10
Let \(f(x)\) be a function measured in pounds, where \(x\) is measured in feet. What are the units of \(f^{\prime \prime}(x) ?\)
Problem 11
Compute the derivative of the given function. $$f(x)=\left(\ln x+x^{2}\right)^{3}$$
Problem 11
Use the definition of the derivative to compute the derivative of the given function. $$h(x)=x^{3}$$
Problem 11
Let \(v(t)\) measure the velocity, in \(\mathrm{ft} / \mathrm{s}\), of a car moving in a straight line \(t\) seconds after starting. What are the units of \(v^{\prime}(t) ?\)
Problem 11
(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. $$f(x)=\frac{x^{2}+3}{x}$$
Problem 11
Compute the derivative of the given function. $$g(x)=\frac{x+7}{\sqrt{x}}$$
Problem 11
Compute the derivative of the given function. $$f(x)=7 x^{2}-5 x+7$$
Problem 12
(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. $$g(x)=\frac{x^{3}-2 x^{2}}{2 x^{2}}$$
Problem 12
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 \(\left(f^{-1}\right)^{\prime}(8)\)
