91Ó°ÊÓ

Suppose that the radius \(r\) and surface area \(S=4 \pi r^{2}\) of a sphere are differentiable functions of \(t .\) Write an equation that relates \(d S / d t\) to \(d r / d t\).

a. Find \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) together. c. Evaluate \(d f / d x\) at \(x=a\) and \(d f^{-1} / d x\) at \(x=f(a)\) to show that at these points \(d f^{-1} / d x=1 /(d f / d x)\). $$f(x)=(1 / 5) x+7, \quad a=-1$$

Using the definition, calculate the derivatives of the functions in Exercises \(1-6 .\) Then find the values of the derivatives as specified. $$F(x)=(x-1)^{2}+1 ; \quad F^{\prime}(-1), F^{\prime}(0), F^{\prime}(2)$$

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$y=2 u^{3}, \quad u=8 x-1$$

Use implicit differentiation to find \(d y / d x\). $$x^{3}+y^{3}=18 x y$$

Find the first and second derivatives. $$y=x^{2}+x+8$$

Find the linearization \(L(x)\) of \(f(x)\) at \(x=a\). $$f(x)=\sqrt{x^{2}+9}, \quad a=-4$$

Find \(d y / d x\). $$y=\frac{3}{x}+5 \sin x$$

a. Find \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) together. c. Evaluate \(d f / d x\) at \(x=a\) and \(d f^{-1} / d x\) at \(x=f(a)\) to show that at these points \(d f^{-1} / d x=1 /(d f / d x)\). $$f(x)=5-4 x, \quad a=1 / 2$$

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$y=\sin u, \quad u=3 x+1$$