# Calculus 2 : Derivatives of Parametrics

## Example Questions

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### Example Question #1 : Derivatives Of Parametric, Polar, And Vector Functions

Find the derivative of the following set of parametric equations:

Explanation:

We start by taking the derivative of x and y with respect to t, as both of the equations are only in terms of this variable:

The problem asks us to find the derivative of the parametric equations, dy/dx, and we can see from the work below that the dt term is cancelled when we divide dy/dt by dx/dt, leaving us with dy/dx:

So now that we know dx/dt and dy/dt, all we must do to find the derivative of our parametric equations is divide dy/dt by dx/dt:

### Example Question #1 : Derivatives Of Parametrics

Solve:

Explanation:

The integration involves breaking up a power of a trigonometric ratio, and then using known trigonometric identities.

The alternative is to find which answer choice has a derivative equal to the answer choice, and for this we get:

### Example Question #2 : Derivatives Of Parametrics

Solve for  if  and .

Explanation:

Write the the formula to solve for the derivative of parametric functions.

Find and  using the power rule .

Substitute back to the formula to obtain the derivative.

### Example Question #3 : Derivatives Of Parametrics

Find the derivative of the following parametric function:

Explanation:

The derivative of a parametric function is given by:

where

,

The derivatives were found using the following rules:

Simply divide the derivatives as shown above.

### Example Question #4 : Derivatives Of Parametrics

Solve for  if  and .

None of the above

Explanation:

Given equations for  and  in terms of , we can find the derivative of parametric equations as follows:

, as the  terms will cancel out.

Using the Power Rule

for all  and given  and ,

and .

Therefore,

.

### Example Question #5 : Derivatives Of Parametrics

Find the derivative of the following parametric equation:

Explanation:

The derivative of a parametric equation is given by the following equation:

Solving for the derivative of the equation for y, you get

and for the equation for x, you get

The following rules were used for the derivatives:

### Example Question #6 : Derivatives Of Parametrics

Find  if  and .

Explanation:

Write the formula to find the derivative for parametric equations.

Substitute the knowns into the formula.

### Example Question #7 : Derivatives Of Parametrics

Solve for  if  and .

Explanation:

We can determine that   since the  terms will cancel out in the division process.

Since  and ,  we can use the Power Rule

for all  to derive

and .

Thus:

.

### Example Question #1 : Derivatives Of Parametric, Polar, And Vector Functions

Solve for  if  and .

None of the above

Explanation:

We can determine that   since the  terms will cancel out in the division process.

Since  and , we can use the Power Rule

for all  to derive

and  .

Thus:

.

### Example Question #8 : Derivatives Of Parametrics

Solve for  if  and .

None of the above

Explanation:

We can determine that   since the  terms will cancel out in the division process.

Since  and , we can use the Power Rule

for all  to derive

and   .

Thus:

.

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