91Ó°ÊÓ

For each of the following composite functions, find an inner function \(u=g(x)\) and an outer function \(y=f(u)\) such that \(y=f(g(x)) .\) Then calculate \(dy/dx.\) $$y=\sqrt{7 x-1}$$

Evaluate the derivative of the following functions. $$f(x)=2 x \tan ^{-1} x-\ln \left(1+x^{2}\right)$$

Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find \(\frac{d y}{d x}\). b. Find the slope of the curve at the given point. $$\tan x y=x+y ;(0,0)$$

A rectangle initially has dimensions \(2 \mathrm{cm}\) by \(4 \mathrm{cm}\). All sides begin increasing in length at a rate of \(1 \mathrm{cm} / \mathrm{s}\) At what rate is the area of the rectangle increasing after \(20 \mathrm{s} ?\)

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of \(\sqrt{x}\). $$f(t)=t$$

Carry out the following steps. a. Use implicit differentiation to find \(\frac{d y}{d x}\). b. Find the slope of the curve at the given point. $$x y=7 ;(1,7)$$

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of \(\sqrt{x}\). $$f(x)=5$$

a. Use definition ( 2 ) ( \(p .\) 135 ) to find the slope of the line tangent to the graph of \(f\) at \(P\). b. Determine an equation of the tangent line at \(P\). $$f(x)=2 x+1 ; P(0,1)$$

Evaluate the derivative of the following functions. $$f(y)=\tan ^{-1}\left(2 y^{2}-4\right)$$

For each of the following composite functions, find an inner function \(u=g(x)\) and an outer function \(y=f(u)\) such that \(y=f(g(x)) .\) Then calculate \(dy/dx.\) $$y=e^{4 x^{2}+1}$$