# Calculus 2 : First and Second Derivatives of Functions

## Example Questions

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### Example Question #1 : First And Second Derivatives Of Functions

Find the second derivative of the following equation:

Explanation:

To find the second derivative, first we need to find the first derivative. The derivative of a natural log is the derivative of operand times the inverse of the operand. So for the given function, we get the first derivative to be

Now, we have to take the derivative of the first derivative. To simplify this, we can rewrite the function to be . From here we can use the chain rule to solve for the derivative. First, multiply by the exponent and find the new exponent by subtracting the old one by one. Next multiply by the derivative of (2x-1) and then simplify. Thus, we get

.

### Example Question #2 : First And Second Derivatives Of Functions

Find the second derivative of the following function.

Explanation:

To find the second derivative, first we need to find the first derivative. So for the given function, we get the first derivative to be

Now we have to take the derivative of the derivative. To do this we need to use the product rule as shown below

Thus, we get

.

### Example Question #3 : First And Second Derivatives Of Functions

Find the second derivative of the given function:

Explanation:

To find the second derivative, first we need to find the first derivative. To find the first derivative we need to use the quotient rule as follows. So for the given function, we get the first derivative to be

Now we have to take the derivative of the derivative. To do this we need to use the quotient rule as shown below.

Thus, we get

### Example Question #4 : First And Second Derivatives Of Functions

Calculate

Explanation:

There are two seprate functions that make up . There is  and .

On a general note,

and

.

Also,

With that said, let's calculate :

. Notice the  term is still unchanged.

And now let's calculate .

### Example Question #5 : First And Second Derivatives Of Functions

Find  and .

,

Explanation:

To find a and b, first let's calculate .

Remember that , where a is real number.

is simply the coefficient in front of the exponential, which simplifies to 1, and  is the power of the exponent, which is 2.

### Example Question #6 : First And Second Derivatives Of Functions

Determine the derivative of  with the respect to .

Explanation:

In order so solve the derivative with the respect to x, implicit differentiation is required.  The notation for finding the derivative of the function with the respect to x is:

Take the derivative and apply chain rule where necessary.

### Example Question #7 : First And Second Derivatives Of Functions

Find the derivative of .

Explanation:

To solve this derivative, we need to use logarithmic differentiation. This allows us to use the logarithm rule  to solve an easier derivative.

Let .

Now we'll take the natural log of both sides to get

.

Now we can use implicit differentiation to solve for .

The derivative of  is , and the derivative of  can be found using the product rule, which states

where  and  are functions of .

Letting  and

(which means  and ) we get our derivative to be .

Now we have , but , so subbing that in we get

.

Multiplying both sides by , we get

.

That is our derivative.

### Example Question #8 : First And Second Derivatives Of Functions

Find the first derivative of the function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

### Example Question #9 : First And Second Derivatives Of Functions

Find the second derivative of the following function:

Explanation:

The first derivative of the function is equal to

The second derivative of the function (the derivative of the above function) is

The following rules were used for the derivatives:

### Example Question #10 : First And Second Derivatives Of Functions

The position of a car is given by the following function:

What is the velocity function of the car?