91Ó°ÊÓ

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=x^{1 / x} $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t .\) Assume that \(W(0)=6\) and that $$ \frac{d W}{d t}=-3 W(t) $$ (a) How much material is left at time \(t=4 ?\) (b) What is the half-life of the material?

Find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=\frac{x^{2}+3}{x^{3}+5}\), at \(x=0\)

Find a point on the curve $$ y=4-x^{2} $$ whose tangent line is parallel to the line \(y=2\). Is there more than one such point? If so, find all other points with this property.

Find the derivatives of the following functions: $$ f(x)=\sin 2 x+\sin ^{2} x $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t .\) Assume that \(W(0)=10\) and \(W(1)=8\). (a) Find the differential equation that describes this situation. (b) How much material is left at time \(t=5\) ? (c) What is the half-life of the material?

Find a point on the curve $$ y=(4-x)^{2} $$ whose tangent line is parallel to the line \(y=-3\). Is there more than one such point? If so, find all other points with this property.

Find the derivatives of the following functions: $$ f(x)=\sec ^{2}\left(2 x^{2}-1\right) $$

Find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=\frac{1}{x}-\frac{2}{\sqrt{x}}+\frac{4}{x^{2}}\), at \(x=1\)

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=x^{3 / x} $$