Find derivatives for the functions. Assume a, b, c, and k are constants.g(x)=5x^{4}

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 derivatives of the functions. ${\displaystyle f\left(x\right)=e^x\ln \left|x\right|}$

Calculate the second-order partial derivatives off ${\displaystyle (x,y)=x^3+y^2{e}^x}$

On a frictionless, horizontal air table, puck A (with mass 0.250 kg) is moving toward puck B (with mass 0.350 kg), which is initially at rest. After the collision, puck A has a velocity of 0.120 m/s to the left, and puck B has a velocity of 0.650 m/s to the right.
(a) What was the speed of puck A before the collision?
(b) Calculate the change in the total kinetic energy of the system that occurs during the collision.

Find the derivatives of the given functions. ${\displaystyle w\left(x\right)=\sec x\tan (x^2-1)}$

Find both first partial derivatives. ${\displaystyle z=e^{xy}}$