derivatives math question

Question

Assume A and k are constants. Compute the following derivatives. All answers should be written in terms of t, A and k.

1.d/dt (Asin(t) + cos(kt)) = ________

2.d/dt (Asin(t)cos(kt)) = _______

3.d/dt (Ae^-kt cos(t)) = _________

Answer 1

Framework for the solutions:

u(t) is some function of t

d/dt[ A * u(t) ] = A * u'(t)

d/dt[ cos( u(t) ) ] = - sin[ u(t) ] * u'(t)

d/dt[ sin( u(t) ) ] = cos[ u(t) ] * u'(t)

d/dt[ e^u(t) ] = e^u(t) * u'(t)

f(t), g(t), u(t), and v(t) are all some functions of t

d/dt[ f( u(t) ) * g( v(t) ) ] = f( u(t) ) * g'( v(t) ) * v'(t) + g( v(t) ) * f'( u(t) ) * u'(t)

d/dt[ f( u(t) ) + g( v(t) ) ] = f'( u(t) ) * u'(t) + g'( v(t) ) * v'(t)

∴ d/dt[ A sin t + cos kt ] = A cos t - k sin kt

∴ d/dt[ A sin t cos kt ] = -kA sin t sin kt + A cos t cos kt

∴ d/dt[ A e^-kt cos t ] = - A e^-kt sin t - kA e^-kt cos t

I hope this helps! Message me in the comments with any questions, comments, or concerns!