Chapter 3: Q18E (page 173)
18. Find \(\frac{{dy}}{{dx}}\) by implicit differentiation.
18. \(\sin x\cos y = {x^2} - 5y\)
The value is \(\frac{{dy}}{{dx}} = \frac{{2x - \cos x\cos y}}{{5 - \sin x\sin y}}\).
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53-56 Find \(y'\) and \(y''\).
54. \(y = {\left( {{\bf{1}} + \sqrt x } \right)^{\bf{3}}}\)
Find \(f'\left( x \right)\) and \(f''\left( x \right)\).
34.\(f\left( x \right) = \frac{x}{{1 + \sqrt x }}\)
The Michaelis-Menten equation fir the enzyme chymotrypsin is
\[v = \frac{{{\bf{0}}{\bf{.14}}\left[ S \right]}}{{{\bf{0}}.{\bf{015}} + \left[ S \right]}}\]
where v is the rate of an enzymatic reaction and [S] is the concentration of substrate S. Calculate \[\frac{{{\bf{d}}v}}{{{\bf{d}}\left[ S \right]}}\] and interpret it.
Find the derivative of the function.
33. \(F\left( t \right) = {e^{t\sin 2t}}\)
Write the composite function in the form \(f\left( {g\left( x \right)} \right)\). (Identify the inner function \(u = g\left( x \right)\) and the outer function \(y = f\left( u \right)\).) Then find the derivative \(\frac{{dy}}{{dx}}\).
5. \(y = {e^{\sqrt x }}\)
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