91Ó°ÊÓ

Find \(d y / d x\). $$ y \equiv \sin ^{-1} x+\cos ^{-1} x $$

Find the limit by interpreting the expression as an appropriate derivative. (a) \(\lim _{x \rightarrow 0} \frac{\ln (1+3 x)}{x}\) (b) \(\lim _{x \rightarrow 0} \frac{\ln (1-5 x)}{x}\)

Use the differential \(d y\) to approximate \(\Delta y\) when \(x\) changes as indicated. $$ y=\frac{x}{x^{2}+1} ; \text { from } x=2 \text { to } x=1.96 $$

Make a conjecture about the equations of horizontal asymptotes, if any, by graphing the equation with a graphing utility; then check your answer using L'Hôpital's rule. $$ y=\ln x-e^{x} $$

Find \(d y / d x\). $$ \underline{y} \equiv \underline{x}^{2}\left(\sin ^{-1} x\right)^{3} $$

Make a conjecture about the equations of horizontal asymptotes, if any, by graphing the equation with a graphing utility; then check your answer using L'Hôpital's rule. $$ y=x-\ln \left(1+2 e^{x}\right) $$

Find the limit by interpreting the expression as an appropriate derivative. (a) \(\lim _{\Delta x \rightarrow 0} \frac{\ln \left(e^{2}+\Delta x\right)-2}{\Delta x}\) (b) \(\lim _{w \rightarrow 1} \frac{\ln w}{w-1}\)

Use the differential \(d y\) to approximate \(\Delta y\) when \(x\) changes as indicated. $$ y=x \sqrt{8 x+1} ; \text { from } x=3 \text { to } x=3.05 $$

Find \(d y / d x\). $$ y=\sec ^{-1} x+\csc ^{-1} x $$

Find the limit by interpreting the expression as an appropriate derivative. (a) \(\lim _{x \rightarrow 0} \frac{\ln (\cos x)}{x}\) (b) \(\lim _{h \rightarrow 0} \frac{(1+h)^{\sqrt{2}}-1}{h}\)