Find the derivatives of \(\displaystyle f{{\left({x}\right)}}=\frac{3}{{2}}{x}^{4}+{2}{x}^{3}-{5}{x}^{2}+{9}\)

Find the derivatives of \(\displaystyle{h}{\left({x}\right)}={\left(-{7}{\left({x}^{2}-{8}\right)}{2}\right)}{2}{x}\)

Need help with the following question Compute the directional derivative of \(\displaystyle f{{\left({x},{y}\right)}}={e}^{2}{x}-{5}{y}\)

Define \(\displaystyle g{{\left({x},{y}\right)}}={x}^{2}+{y}^{2}-{4}{x}{y}+{3}{y}+{2}\)

Find the derivative \(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\arctan{{\left({x}^{{2}}+{1}\right)}}}\right]}\) a) \(\displaystyle{\left({a}{r}{c}{\sec{{\left({x}^{{2}}+{1}\right)}}}\right)}^{{2}}\) b) \(\displaystyle{\frac{{{1}}}{{{2}+{x}^{{2}}}}}\) c) \(\displaystyle{\frac{{{1}}}{{{1}+{\left({x}^{{2}}+{1}\right)}^{{2}}}}}\) d) \(\displaystyle{\frac{{{2}{x}}}{{{1}+{\left({x}^{{2}}+{1}\right)}^{{2}}}}}\)

Find the point on the line \(y=4x+3\) that is closest to the origin. (x,y)=( )

\({y}{\left({x}^{2}-{1}\right)}{\left.{d}{y}\right.}+{x}{\left({y}^{2}+{1}\right)}{\left.{d}{x}\right.}={0}\)

Find \(\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\) and \(\displaystyle\frac{{{d}^{{{2}}}y}}{{\left.{d}{x}\right.}^{{{2}}}}\). \(\displaystyle{x}={e}^{{{t}}},{y}={t}{e}^{{−{t}}}\) For which values of t is the curve concave upward?

Check whether the given function is satisfying the Fubini’s theorem about mixed partial derivative \(\displaystyle f{{\left({x},{y}\right)}}={3}{x}^{2} \sin{{\left({4}{x}{y}\right)}}\)

how to find the derivative of \(\displaystyle f{{\left({u}\right)}}={9}{e}^{u}+{20}\)

Explain why neither Substitution nor Integration by Parts would work to evaluate integral of \(\displaystyle \cos{{\left({x}^{2}\right)}}{\left.{d}{x}\right.}\)