derivatives calculus

Question

Calculate the following derivatives. Assume that any letter other than t is a constant.

d/dt (h₀ + v₀t + 1/2gt^2) = __________

d/dt (Ce^-kt) = ______________

d/dt (P₀ + k/t^2) = ____________

d/dt (at^3 + be^0.1t) = ______________

Answer 1

d/dt(ho + vot + (1/2)gt^2) = vo +gt

this is a formula for h'(t), with g= -32 feet per second per second, vo= initial velocity, ho = initial height measured in feet, t= time measured in seconds or in the metric system g=-9.8 meters per second per second

take the derivative and set equal to zero to solve for time of maximum height

vo +gt = 0

t = -vo/g = vo/32 seconds. IF vo=0, then max height is when t=0, and max h= ho

max height = h(vo/32)= ho + (vo^2)/32 + (1/2)g(vo/32)^2

d/dt( Ce^-kt) = -kCe^-kt

this is a formula for continuously compound interest, with negative interest , depreciation or decay rate

It's the limit of the interest formula A=P(1+r/n)^nt as n approaches infinity

d/dt(Po + k/t^2) = -2k/t^3

d/dt(at^3 + be^0.1t) = 3at^2 + 0.1be^0.1t

Answer 2

Assuming that any letter other than t is a constant, the following derivatives are:

d/dt (h₀ + v₀t + 1/2gt^2) = ___0 + v₀ + g/2 * 2t___ = ___v₀ + gt___

d/dt (Ce^-kt) = ___Ce^(-kt) * (-k)___ = ___-kCe^(-kt)___

d/dt (P₀ + k/t^2) = ___0 + k*(-1/t)___ = ___-k/t___

d/dt (at^3 + be^0.1t) = ___a*(3t^2) + b*0.1*e^(0.1t)___ = ___3at^2 + 0.1be^(0.1t)___