91Ó°ÊÓ

A function \(f\) and a value \(c\) are given. Find an equation of the tangent line to the graph of \(f\) at \((c, f(c))\). $$ f(x)=\sin (x), c=\pi / 3 $$

Differentiate the given expression with respect to \(x\). $$ \cos (\arctan (x)) $$

Use the Inverse Function Derivative Rule to calculate \(\left(f^{-1}\right)^{\prime}(t)\). $$ f:(1, \infty) \rightarrow(2, \infty), f(s)=\log _{3}\left(6+3^{s}\right) $$

Use the Chain Rule-Power Rule to differentiate the given expression with respect to \(x\). $$ (x-1 / x)^{5 / 2} $$

Find the slope of the tangent line to the graph of the given function at the given point \(P\). $$ f(x)=-4 x^{2}+x+1 \quad P=(2,-13) $$

Use the Reciprocal Rule to compute the derivative of the given expression with respect to \(x\) $$ 5 /(\cos (x)+\sin (x)) $$

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(2^{x} \cos (3 x)\)

Use implicit differentiation to find the tangent line to the given curve at the given point \(P_{0}\). \(x e^{x-1}-\ln (x y)=1 \quad P_{0}=(1,1)\)

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=\sqrt{x} /(1+\sqrt{x}), c=9, x=8.6 $$

Use the Inverse Function Derivative Rule to calculate \(\left(f^{-1}\right)^{\prime}(t)\). $$ f:(-\infty, \infty) \rightarrow(0, \infty), f(s)=2^{s} $$