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Example Questions
Example Question #1 : Using Implicit Differentiation
An ellipse is represented by the following equation:
What is the slope of the curve at the point (3,2)?
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It would be difficult to differentiate this equation by isolating . Luckily, we don't have to. Use to represent the derivative of with respect to and follow the chain rule.
(Remember, is the derivative of with respect to , although it usually doesn't get written out because it is equal to 1. We'll write it out this time so you can see how implicit differentiation works.)
Now we need to isolate by first putting all of these terms on the same side:
This is the equation for the derivative at any point on the curve. By substituting in (3, 2) from the original question, we can find the slope at that particular point:
Example Question #1 : Specific Derivatives
Find the derivative for
The derivative must be computed using the product rule. Because the derivative of brings a down as a coefficient, it can be combined with to give
Example Question #2 : Specific Derivatives
Give the instantaneous rate of change of the function at .
The instantaneous rate of change of at is , so we will find and evaluate it at .
for any positive , so
Example Question #3 : Specific Derivatives
What is ?
Therefore,
for any real , so , and
Example Question #4 : Specific Derivatives
What is ?
Therefore,
for any positive , so , and
Example Question #1 : Understanding Derivatives Of Trigonometric Functions
Find the derivative of the following function:
The derivative of is. It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative).
Our function
the factor of 3 does not change when we differentiate, therefore the answer is
Example Question #4 : Specific Derivatives
The derivative of a sine function does not follow the power rule. It is one that should be memorized.
.
Example Question #5 : Specific Derivatives
What is the second derivative of ?
The derivatives of trig functions must be memorized. The first derivative is:
.
To find the second derivative, we take the derivative of our result.
.
Therefore, the second derivative will be .
Example Question #6 : Specific Derivatives
Compute the derivative of the function .
Use the Chain Rule.
Set and substitute.
Example Question #1 : Understanding Derivatives Of Sums, Quotients, And Products
Find the derivative of the following function:
Since this function is a polynomial, we take the derivative of each term separately.
From the power rule, the derivative of
is simply
We can rewrite as
and using the power rule again, we get a derivative of
or
So the answer is
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