Calculus 2 : First and Second Derivatives of Functions

Example Questions

Example Question #101 : 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:

Note that the radical acts as the "outer" function using the first rule, the chain rule.

Example Question #102 : 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 #103 : 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 #104 : 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 #105 : 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 #106 : 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 #107 : First And Second Derivatives Of Functions

Find the second derivative of

Explanation:

To find the second derivative of a function, we must first find the first derivative.

Now, we must differentiate this function.  The first part is a straightforward trigonometric derivative.  For the second term, we must use the chain rule.

Example Question #108 : First And Second Derivatives Of Functions

Find the derivative of .

Explanation:

The derivative of  is special because it returns back the original function .

Therefore, this derivative is simply a chain rule application of that exponential function.

Recall the chain rule is,

Applying this rule to the original function results in the following derivative.

Example Question #109 : First And Second Derivatives Of Functions

Find the second derivative of

Explanation:

To get to the second derivative, we must first take the first derivative!

Then, we will differentiate that to get our answer.

Recall the trigonometric derivatives:

Example Question #110 : First And Second Derivatives Of Functions

What is the derivative of ?