Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!

To take the derivative of the position function

We'll need to make use of the following derivative rule(s):

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

Using the above properties, the velocity function is

To find when the particle is stationary, set this function equal to zero:

Of the answer choices,

All satisfy this condition.

Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!

To take the derivative of the position function

We'll need to make use of the following derivative rule(s):

Derivative of an exponential: 

Trigonometric derivative: 

Product rule: 

Note that u and v may represent large functions, and not just individual variables!

Using the above properties, the velocity function is

At time 

Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!

To take the derivative of the position function

We'll need to make use of the following derivative rule(s):

Derivative of an exponential: 

Derivative of a natural log: 

Trigonometric derivative: 

Note that u and v may represent large functions, and not just individual variables!

Using the above properties, the velocity function is

At time 

Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!

To take the derivative of the position function

We'll need to make use of the following derivative rule(s):

Derivative of an exponential: 

Trigonometric derivative: 

Note that u and v may represent large functions, and not just individual variables!

Using the above properties, the velocity function is

At time 

Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!

To take the derivative of the position function

We'll need to make use of the following derivative rule(s):

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

Using the above properties, the velocity function is

At time 

Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!

To take the derivative of the position function

We'll need to make use of the following derivative rule(s):

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

Using the above properties, the velocity function is

At time 

Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!

To take the derivative of the position function

We'll need to make use of the following derivative rule(s):

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

Using the above properties, the velocity function is

At time 

We can find the velocity of the spaceship over a time frame by taking the derivative of the position equation.  The derivative of the position equation is:

The question is asking what is the instantaneous velocity of the spaceship at the  mark. When we insert  into the velocity equation, we get .  

From the power rule, 

we take the derivative of the position equation 

To find the velocity of the rock, we take the derivative of the position equation.  The velocity equation is

From the velocity equation, we insert  to find the instantaneous velocity.