91Ó°ÊÓ

Find the slope of the graph of the function at the indicated point. Use the derivative feature of a graphing utility to confirm your results. $$ f(x)=3(5-x)^{2} $$ $$ (5,0) $$

Find the derivative of the algebraic function. $$ f(x)=x^{4}\left(1-\frac{2}{x+1}\right) $$

Fixed Point In Exercises 23-26, approximate the fixed point of the function to two decimal places. [A fixed point \(x_{0}\) of a function \(f\) is a value of \(x\) such that \(\left.f\left(x_{0}\right)=x_{0} .\right]\) $$ f(x)=-\ln x $$

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. Function \(\quad\) Line \(f(x)=\frac{1}{\sqrt{x-1}} \quad x+2 y+7=0\)

In Exercises 15-28, find the derivative of the function. $$ y=x \arctan 2 x-\frac{1}{4} \ln \left(1+4 x^{2}\right) $$

Find the slope of the graph of the function at the indicated point. Use the derivative feature of a graphing utility to confirm your results. $$ f(\theta)=4 \sin \theta-\theta $$ $$ (0,0) $$

Find the derivative of the algebraic function. $$ f(x)=\frac{2 x+5}{\sqrt{x}} $$

Use Newton's Method to show that the equation \(x_{n+1}=x_{n}\left(2-a x_{n}\right)\) can be used to approximate \(1 / a\) if \(x_{1}\) is an initial guess for the reciprocal of \(a .\) Note that this method of approximating reciprocals uses only the operations of multiplication and subtraction. [Hint: Consider \(f(x)=(1 / x)-a\).]

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the indicated point. $$ y^{2}=\frac{x^{2}-9}{x^{2}+9} $$

In Exercises 15-28, find the derivative of the function. $$ y=8 \arcsin \frac{x}{4}-\frac{x \sqrt{16-x^{2}}}{2} $$