If a ball is throwndirectly upward with a velocity of 40ft/s, its height (in feet)after t seconds is given by y=40t-16t^2. What is the maximum heightattained by the ball?

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders? Give examples.

Use the given graph to estimate the value of each derivative.(Round all answers to one decimal place.)Graph uploaded below.
(a) f ' (0)1
(b) f ' (1)2
(c) f ' (2)3
(d) f ' (3)4
(e) f ' (4)5
(f) f ' (5)6

Find the first and second derivatives of the functions ${\displaystyle u=\frac{(x^2+x)(x^2-x+1)}{x^4}}$

Prove that ${\displaystyle \underset{x\to 0}{lim}\left[\ln \left(\frac{1+\cos x}{1-\cos x}\right)\right]\sim 2\ln \left|x\right|}$

Find the second order derivatives of the function ${\displaystyle {\cos }^{3}x}$

Find the derivative of y with respect to
x for y =17
$y={17}^x$ is........?

Find the following derivatives.
${\displaystyle \ln \left(\frac{x+3}{x^2}\right)}$