# Calculus 2 : Derivative at a Point

## Example Questions

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### Example Question #1 : Derivative At A Point

Given the function , find the slope of the point .

The slope cannot be determined.

Explanation:

To find the slope at a point of a function, take the derivative of the function.

The derivative of  is .

Therefore the derivative becomes,

since .

Now we substitute the given point to find the slope at that point.

### Example Question #2 : Derivative At A Point

Find the value of the following derivative at the point  :

Explanation:

To solve this problem, first we need to take the derivative of the function. It will be easier to rewrite the equation as  from here we can take the derivative and simplify to get

From here we need to evaluate at the given point . In this case, only the x value is important, so we evaluate our derivative at x=2 to get.

### Example Question #3 : Derivative At A Point

Evaluate the value of the derivative of the given function at the point :

Explanation:

To solve this problem, first we need to take the derivative of the function.

From here we need to evaluate at the given point . In this case, only the x value is important, so we evaluate our derivative at x=1 to get

.

### Example Question #4 : Derivative At A Point

Find the value of the derivate of the given function at the point :

Explanation:

To solve this problem, first we need to take the derivative of the function. To do this we need to use the quotient rule and simplified as follows:

From here we need to evaluate at the given point . In this case, only the x value is important, so we evaluate our derivative at x=2 to get

.

### Example Question #5 : Derivative At A Point

Find the slope at  for the function .

Explanation:

The derivative of  is:

Substitute the point at .

### Example Question #6 : Derivative At A Point

What is the slope of  at ?

Explanation:

In order to find the slope of a function at a certain point, plug in that point into the first derivative of the function. Our first step here is to take the first derivative.

Since we see that f(x) is composed of two different functions, we must use the product rule. Remember that the product rule goes as follows:

Following that procedure, we set  equal to  and  equal to .

,

which can be simplified to

.

Now plug in 1 to find the slope at x=1.

Remember that .

### Example Question #7 : Derivative At A Point

Consider the function:  .  What is the derivative at ?

Explanation:

To solve for the derivative of , use implicit differentiation which means to take the derivative of each term with respect to the variable in that term.

Substitute the point into the derivative.

### Example Question #8 : Derivative At A Point

Use implicit differentiation to find the slope of the tangent line to  at the point .

Explanation:

We must take the derivative  because that will give us the slope. On the left side we'll get

, and on the right side we'll get .

We include the  on the left side because  is a function of , so its derivative is unknown (hence we are trying to solve for it!).

Now we can factor out a  on the left side to get

and divide by  in order to solve for .

Doing this gives you

.

We want to find the slope at , so we can sub in  for  and

.

### Example Question #9 : Derivative At A Point

Find the derivative of the following function at the point .

Explanation:

Here, we must use the product rule.

Assuming  and , our expression becomes

.

The question asks us to evaluate this at .

### Example Question #1 : Derivative At A Point

Given the function , what is the slope at the point ?