91Ó°ÊÓ

Find the second derivative of the function \(y=x^{2} \ln x\).

Solve the given problems. Display the graphs of \(y_{1}=\cos x\) and \(y_{n}=\frac{\sin (x+h)-\sin x}{h}\) on the same screen of a calculator for \(0 \leq x \leq 2 \pi .\) For \(n=2,\) let \(h=0.5 ;\) for \(n=3,\) let \(h=0.1 .\) (You might try some even smaller values of \(h .\) ) Use a heavier curve for \(y_{3} .\) What do these curves show?

Solve the given problems. Find the second derivative of the function \(y=x^{2} \ln x\)

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.

Solve the given problems. Evaluate the derivative of \(y=\sin ^{-1} 2 x+\sqrt{1-4 x^{2}},\) where \(x=0.250\)

Solve the given problems by finding the appropriate derivative. The relative number \(N\) of gas molecules in a container that are moving at a velocity \(v\) can be shown to be \(N=a v^{2} e^{-b v^{2}},\) where \(a\) and \(b\) are constants. Find \(v\) for the maximum \(N\).

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

Evaluate the derivative of \(y=\sin ^{-1} 2 x+\sqrt{1-4 x^{2}},\) where \(x=0.250\).

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

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.