91Ó°ÊÓ

Finding an Equation of a Tangent Line In Exercises \(65-72,\) (a) find an equation of the tangent line to the graph of \(f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ y=\ln x^{4}, \quad(1,0) $$

In Exercises 65–74, find the derivative $$ y=\cosh ^{-1}(3 x) $$

Use a computer algebra system to find the linear approximation \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and the quadratic approximation \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) of the function \(f\) at \(x=a\) . Sketch the graph of the function and its linear and quadratic approximations. \(f(x)=\arctan x, \quad a=0\)

Logarithmic Differentiation In Exercises \(63-66,\) use logarithmic differentiation to find \(d y / d x .\) $$ y=(x-2)^{x+1} $$

Finding the Equation of a Tangent Line In Exercises 65 and \(66,\) find an equation of the tangent line to the graph of the function at the given point. $$ x e^{y}+y e^{x}=1, \quad(0,1) $$

Verify that \(f\) has an inverse. Then use the function \(f\) and the given real number \(a\) to find \(\left(f^{-1}\right)^{\prime}(a) .\) (Hint: See Example \(5 . )\) \(f(x)=\frac{1}{27}\left(x^{5}+2 x^{3}\right), \quad a=-11\)

Verify that \(f\) has an inverse. Then use the function \(f\) and the given real number \(a\) to find \(\left(f^{-1}\right)^{\prime}(a) .\) (Hint: See Example \(5 . )\) \(f(x)=\sqrt{x-4}, \quad a=2\)

Logarithmic Differentiation In Exercises \(63-66,\) use logarithmic differentiation to find \(d y / d x .\) $$ y=(1+x)^{1 / x} $$

Finding the Equation of a Tangent Line In Exercises 65 and \(66,\) find an equation of the tangent line to the graph of the function at the given point. $$ 1+\ln x y=e^{x-y}, \quad(1,1) $$

In Exercises 65–74, find the derivative $$ y=\tanh ^{-1} \frac{x}{2} $$