91Ó°ÊÓ

$$ \text { } , \text { find the indicated derivative. } $$$$ D_{x} \ln (x-4)^{3} $$

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{3}{x^{3}}+x^{-4} $$

If \(y=x^{4}+2 x\), find the values of \(\Delta y\) and \(d y\) in each case. (a) \(x=2\) and \(d x=\Delta x=1\) (b) \(x=2\) and \(d x=\Delta x=0.005\)

$$ \text { } , \text { find the indicated derivative. } $$ $$ D_{x} \ln \sqrt{3 x-2} $$

Find \(D_{x} y\). $$ y=\left(2-3 x^{2}\right)^{4}\left(x^{7}+3\right)^{3} $$

A business is prospering in such a way that its total (accumulated) profit after \(t\) years is \(1000 t^{2}\) dollars. (a) How much did the business make during the third year (between \(t=2\) and \(t=3) ?\) (b) What was its average rate of profit during the first half of the third year, between \(t=2\) and \(t=2.5 ?\) (The rate will be in dollars per year.) (c) What was its instantaneous rate of profit at \(t=2 ?\)

Find \(D_{x} y\) using the rules of this section. $$ y=2 x^{-6}+x^{-1} $$

Find a formula for $$ D_{x}^{n}\left(a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\right) $$

The vertex angle \(\theta\) opposite the base of an isosceles triangle with equal sides of length 100 centimeters is increasing at \(\frac{1}{10}\) radian per minute. How fast is the area of the triangle increasing when the vertex angle measures \(\pi / 6\) radians? Hint: \(A=\frac{1}{2} a b \sin \theta\)

$$ \text { } \text { find } D_{x} y . $$ $$ y=\sec ^{3} x $$