We have the equation #p(t)=t-tsin(pi/4t)#
Since the derivative of position is velocity, or #p'(t)=v(t)#, we must calculate:
#d/dt(t-tsin(pi/4t))#
According to the difference rule, we can write:
#d/dtt-d/dt(tsin(pi/4t))#
Since #d/dtt=1#, this means:
#1-d/dt(tsin(pi/4t))#
According to the product rule, #(f*g)'=f'g+fg'#.
Here, #f=t# and #g=sin((pit)/4)#
#1-(d/dtt*sin((pit)/4)+t*d/dt(sin((pit)/4)))#
#1-(1*sin((pit)/4)+t*d/dt(sin((pit)/4)))#
We must solve for #d/dt(sin((pit)/4))#
Use the chain rule:
#d/dxsin(x)*d/dt((pit)/4)#, where #x=(pit)/4#.
#=cos(x)*pi/4#
#=cos((pit)/4)pi/4#
Now we have:
#1-(sin((pit)/4)+cos((pit)/4)pi/4t)#
#1-(sin((pit)/4)+(pitcos((pit)/4))/4)#
#1-sin((pit)/4)-(pitcos((pit)/4))/4#
That's #v(t)#.
So #v(t)=1-sin((pit)/4)-(pitcos((pit)/4))/4#
Therefore, #v(7)=1-sin((7pi)/4)-(7picos((7pi)/4))/4#
#v(7)=-2.18"m/s"#, or #2.18"m/s"# in terms of speed.
