# Calculus 2 : First and Second Derivatives of Functions

## Example Questions

### Example Question #11 : First And Second Derivatives Of Functions

Find the derivative of the following function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

### Example Question #1 : Velocity, Speed, Acceleration

Let

Find the first and second derivative of the function.

Explanation:

In order to solve for the first and second derivative, we must use the chain rule.

The chain rule states that if

and

then the derivative is

In order to find the first derviative of the function

we set

and

Because the derivative of the exponential function is the exponential function itself, we get

And differentiating  we use the power rule which states

As such

And so

To solve for the second derivative we set

and

Because the derivative of the exponential function is the exponential function itself, we get

And differentiating  we use the power rule which states

As such

And so the second derivative becomes

### Example Question #12 : First And Second Derivatives Of Functions

Find the derivative of the following function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

### Example Question #13 : First And Second Derivatives Of Functions

Find the derivative of the function:

Explanation:

The integral of the function is equal to

and was found using the following rules:

Note that it is easier to integrate the first term once it is rewritten as .

### Example Question #3 : Velocity, Speed, Acceleration

Find the velocity function of the particle if its position is given by the following function:

Explanation:

The velocity function is given by the first derivative of the position function:

and was found using the following rules:

### Example Question #31 : Derivatives

Find the second derivative of the following function:

Explanation:

The first derivative of the function is equal to

The second derivative - the derivative of the function above - of the original function is equal to

Both derivatives were found using the following rules:

### Example Question #15 : First And Second Derivatives Of Functions

Find the derivative of the following function:

Explanation:

The derivative was found using the following rules:

### Example Question #16 : First And Second Derivatives Of Functions

Find the derivative of the function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

### Example Question #17 : First And Second Derivatives Of Functions

Find the derivative of the following function:

Explanation:

The derivative of the function was found using the following rules:

### Example Question #18 : First And Second Derivatives Of Functions

Solve for the derivative: