Question

sketch proof of the product rule for a scalar function of parameter t and a vector function of parameter t ?(dont know how to get notation into the question)

Answer 1

\[d(\lambda f)/dt=\lambda d(f)t/dt +f d(\lambda)/dt\]
where lambda is the scalar function from reals to reals and f is the vector function from reals to 3 dimensional space (in this example)

Answer 2

first step is to scale the function thru the vector

Answer 3

f(t)<x(t),y(t),z(t)> = <f(t)x(t),f(t)y(t),f(t)z(t)>

now take the derivative of the vector function as usual; componentwise; its just that thyis time each component IS a product

<f'(t)x(t)+f(t)x'(t), f'(t)y(t)+f(t)y'(t), f'(t)z(t)+f(t)z'(t)>

notice that the vector produced is a sum of 2 vectors ... so lets split this into its 2 vectors by removing everything with a f'(t).

<f(t)x'(t), f(t)y'(t), f(t)z'(t)> + <f'(t)x(t), f'(t)y(t), f'(t)z(t)>

factor out the scalars:

f(t)<x'(t), y'(t), z'(t)> + f'(t)<x(t), y(t), f'(t)>

and we are done