Question

Differentiate 7 sin x cos y=2 with respect to t (x and y are functions of t)

Answer 1

d/dt(7 sin(x(t)) cos(y(t))) = d/dt(2)
Factor out constants:
= 7 (d/dt(sin(x(t)) cos(y(t)))) = d/dt(2)
Use the product rule, d/dt(u v) = v ( du)/( dt)+u ( dv)/( dt), where u = cos(y(t)) and v = sin(x(t)):
= 7 (cos(y(t)) (d/dt(sin(x(t))))+sin(x(t)) (d/dt(cos(y(t))))) = d/dt(2)
Use the chain rule, d/dt(sin(x(t))) = ( dsin(u))/( du) ( du)/( dt), where u = x(t) and ( dsin(u))/( du) = cos(u):
= 7 (cos(y(t)) (cos(x(t)) (d/dt(x(t))))+sin(x(t)) (d/dt(cos(y(t))))) = d/dt(2)
The derivative of x(t) is x'(t):
= 7 (sin(x(t)) (d/dt(cos(y(t))))+x'(t) cos(x(t)) cos(y(t))) = d/dt(2)
Use the chain rule, d/dt(cos(y(t))) = ( dcos(u))/( du) ( du)/( dt), where u = y(t) and ( dcos(u))/( du) = -sin(u):
= 7 (sin(x(t)) (sin(y(t)) (-(d/dt(y(t)))))+x'(t) cos(x(t)) cos(y(t))) = d/dt(2)
The derivative of y(t) is y'(t):
=7 (x'(t) cos(x(t)) cos(y(t))-sin(x(t)) y'(t) sin(y(t))) = d/dt(2)
The derivative of 2 is zero:
= 7 (x'(t) cos(x(t)) cos(y(t))-sin(x(t)) y'(t) sin(y(t))) = 0