# Calculus 2 : Definition of Derivative

## Example Questions

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### Example Question #1 : Derivative Review

Evaluate the limit using one of the definitions of a derivative.

Does not exist

Explanation:

Evaluating the limit directly will produce an indeterminant solution of .

The limit definition of a derivative is . However, the alternative form, , better suits the given limit.

Let  and notice . It follows that .

Thus, the limit is

### Example Question #2 : Derivative Review

Evaluate the limit using one of the definitions of a derivative.

Does not exist

Explanation:

Evaluating the derivative directly will produce an indeterminant solution of .

The limit definition of a derivative is . However, the alternative form, , better suits the given limit.

Let  and notice . It follows that .  Thus, the limit is .

### Example Question #1 : Definition Of Derivative

Suppose  and  are differentiable functions, and . Calculate the derivative of , at

Explanation:

Taking the derivative of  involves the product rule, and the chain rule.

Substituting  into both sides of the derivative we get

.

### Example Question #4 : Derivative Review

Evaluate the limit

without using L'Hopital's rule.

Explanation:

If we recall the definition of a derivative of a function  at a point , one of the definitions is

.

If we compare this definition to the limit

we see that that this is the limit definition of a derivative, so we need to find the function  and the point  at which we are evaluating the derivative at. It is easy to see that the function is  and the point is . So finding the limit above is equivalent to finding .

We know that the derivative is , so we have

.

### Example Question #1 : Derivative Review

Approximate the derivative if  where .

Explanation:

Write the definition of the limit.

Substitute .

Since  is approaching to zero, it would be best to evaluate when we assume that  is progressively decreasing.  Let's assume  and check the pattern.

## Find f'(x):

Explanation:

Computation of the derivative requires the use of the Product Rule and Chain Rule.

The Product Rule is used in a scenario when one has two differentiable functions multiplied by each other:

This can be easily stated in words as: "First times the derivative of the second, plus the second times the derivative of the first."

In the problem statement, we are given:

is the "First" function, and  is the "Second" function.

The "Second" function requires use of the Chain Rule.

When:

Applying these formulas results in:

Simplifying the terms inside the brackets results in:

We notice that there is a common term that can be factored out in the sets of equations on either side of the "+" sign. Let's factor these out, and make the equation look "cleaner".

Inside the brackets, it is possible to clean up the terms into one expanded function. Let us do this:

Simplifying this results in one of the answer choices:

### Example Question #7 : Derivative Review

What is the value of the limit below?

Explanation:

Recall that one definition for the derivative of a function  is .

This means that this question is asking us to find the value of the derivative of  at .

Since

and , the value of the limit is .

### Example Question #8 : Derivative Review

Explanation:

Evaluation of this integral requires use of the Product Rule. One must also need to recall the form of the derivative of .

Product Rule:

Applying these two rules results in:

This matches one of the answer choices.

### Example Question #9 : Derivative Review

Use the definition of the derivative to solve for .

Explanation:

In order to find , we need to remember how to find  by using the definition of derivative.

Definition of Derivative:

Now lets apply this to our problem.

Now lets expand the numerator.

We can simplify this to

Now factor out an h to get

We can simplify and then evaluate the limit.

### Example Question #10 : Derivative Review

Use the definition of the derivative to solve for .

Explanation:

In order to find , we need to remember how to find  by using the definition of derivative.

Definition of Derivative:

Now lets apply this to our problem.

Now lets expand the numerator.

We can simplify this to

Now factor out an h to get

We can simplify and then evaluate the limit.

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