Calculus 2 : First and Second Derivatives of Functions

Example Questions

Example Question #281 : Derivatives

Find the derivative of.

Explanation:

Recall that when taking the derivative, multiply the exponent of the x term by the coefficient in front and then subtract one from the exponent.

.

Example Question #282 : Derivatives

Find the derivative of .

Explanation:

To find the derivative here, you have to use implicit differentiation.

The first step is to take the derivative of each term using the power rule,

in this particular case it would look like this:

.

Then, solve for .

.

Example Question #283 : Derivatives

Find the first derivative of

None of the Above

Explanation:

Step 1: Recall the derivative rules:

-For any term with an exponent, the exponent drops and gets multiplied to the coefficient of that term. The new exponent is one less than the original one..
-For any term in the form "ax", the derivative of this term is just the coefficient.
-For any term with no x term (constant), the derivative is always

Step 2: Take the derivative:

The first derivative of  is

Example Question #284 : Derivatives

Find the first derivative of

Explanation:

Step 1: Re-write the function in terms of a term, not a fraction:

Step 2: Using the power rule, the  drops down and is put in front of the x term. The exponent gets subtracted by , which means  is the new exponent..

Step 3: Re-write the exponents with positive numbers...

When we change the exponents to positive from negative, you flip the exponent term over . We can rewrite  as .

Step 4: Multiply the coefficient from Step  and the fraction in step  to get the derivative:

Example Question #285 : Derivatives

Find the derivative of .

Explanation:

This is a special derivative.

Example Question #286 : Derivatives

What is the derivative of the following function?

.

Explanation:

The derivative of a constant raised to a power is given by the following equation.

Example Question #287 : Derivatives

Find the derivative of

Explanation:

This derivative is an application of the power rule.  Remember, we need to multiply the coefficient by the exponent, then reduce the exponent by one.

Example Question #288 : Derivatives

Find the derivative of the function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

Example Question #281 : Derivative Review

Find the derivative of the function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

Example Question #290 : Derivatives

Find the derivative of the function: