All Calculus 2 Resources
Example Questions
Example Question #31 : First And Second Derivatives Of Functions
Find the derivative of the following function at :
To find the derivative, we must use the following rule:
Now, using the above rule, write out the derivative:
The internal derivatives were found using the following rules:
, , ,
Evaluated at , we get
Example Question #32 : First And Second Derivatives Of Functions
Find the derivative of the function .
None of the other answers
We will need to use the product rule and the chain rule to find the derivative.
. Start
. Product Rule
. Use the Chain Rule for .
. Multiply
. Factor out a , and an .
Example Question #231 : Derivative Review
Find the derivative of the function
None of the other answers.
Although we could use the Product Rule to compute the derivative, it becomes much easier to find if we rewrite .
. Start
.
Example Question #31 : First And Second Derivatives Of Functions
If , find in terms of and .
None of the other answers
Using a combination of logarithms, implicit differentiation, and a bit of algebra, we have
. Quotient Rule + implicit differentiation.
Example Question #32 : First And Second Derivatives Of Functions
Find the derivative of the function
None of the other answers
None of the other answers
The correct answer is .
Using the Quotient Rule and the fact , we have
.
Example Question #31 : First And Second Derivatives Of Functions
Find the derivative of the function
None of the other answers
The Chain Rule is required here.
. Start
The Chain Rule Proceeds as follows: . In this case
and
.
Putting these into the Chain Rule, we get
Or the same to say
.
Example Question #231 : Derivative Review
Evaluate the derivative of , where is any constant.
None of the other answers
For the term, we simply use the power rule to abtain . Since is a constant (not a variable), we treat it as such. The derivative of any constant (or "stand-alone number") is .
Example Question #32 : First And Second Derivatives Of Functions
Find the derivative of the function .
None of the other answers
We use the Product Rule to find our answer here. The Product Rule formula is .
Let , , then we have , .
Putting these into our formula, we have
.
Example Question #1364 : Calculus Ii
What is the derivative of
?
We can find the derivative of
using the power rule
with
so we have
Example Question #1365 : Calculus Ii
Find the velocity function given the displacement function:
The derivative of the displacement function is the velocity, so we need to find . We can use the power rule
with
to get