91Ó°ÊÓ

Solve the given problems. Find the derivative of each member of the identity \(1+\tan ^{2} x=\sec ^{2} x\) and show that the results are equal.

Find the linearization \(L(x)\) for the function \(f(x)=2 \ln \tan x\) for \(a=\pi / 4\).

Solve the given problems. Find the derivative \(\frac{d y}{d x}\) of the implicit function \(\sin (x y)+\cos 2 y=x^{2}\)

Solve the given problems. Show that \(y=x e^{-x}\) satisfies the equation \((d y / d x)+y=e^{-x}\).

Solve the given problems. Find the derivative \(\frac{d y}{d x}\) of the implicit function \(x \cos 2 y+\sin x \cos y=1\)

Solve the given problems. Find the points where a tangent to the curve of \(y=\tan x\) is parallel to the line \(y=2 x\) if \(0

Find the differential of the function \(y=6 \log _{x} 2\).

Solve the given problems by finding the appropriate derivative. A missile is launched and travels along a path that can be represented by \(y=\sqrt{x} .\) A radar tracking station is located \(2.00 \mathrm{km}\) directly behind the launch pad. Placing the launch pad at the origin and the radar station at \((-2.00,0),\) find the largest angle of elevation required of the radar to track the missile.

Use a calculator to display the graphs of \(y=\tan ^{-1} x\) and \(y=1 /\left(1+x^{2}\right) .\) By roughly estimating slopes of tangent lines of \(y=\tan ^{-1} x,\) note that \(y=1 /\left(1+x^{2}\right)\) gives reasonable values for the derivative of \(y=\tan ^{-1} x.\)

Solve the given problems. $$\text { Is } \lim _{x \rightarrow 3} \frac{x^{3}-3 x^{2}+x-3}{x^{2}-9}=\lim _{x \rightarrow 3} \frac{3 x^{2}-6 x+1}{2 x}=\frac{5}{3} ? \text { Explain. }$$