# AP Calculus BC : Derivative Rules for Sums, Products, and Quotients

## Example Questions

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### 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

.

## 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 #1 : Definition Of Derivative

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.

### Compute the derivative:

Explanation:

Computation of this derivative will require the use of the Product Rule, and knowledge of the derivative of the inverse tangent function, and natural logarithmic function:

We can now easily compute the derivative.

This simplifies to:

This is one of the answer choices.

### Find dy/dx:

Explanation:

Solving for the derivative requires knowledge of the rule for the inverse tangent function:

In our case:

We can take the derivative of this using the product rule:

Now we can simply plug all of this into the above formula and we arrive at:

Simplifying this further gives:

### Example Question #81 : Derivatives

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 #81 : Derivatives

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 #1 : Derivative Rules For Sums, Products, And Quotients

Find the slope of the tangent line to the function  at .

Explanation:

The slope of the tangent line to a function at a point is the value of the derivative of the function at that point. In this problem,  is a quotient of two functions, , so the quotient rule is needed.

In general, the quotient rule is

.

To apply the quotient rule in this example, you must also know that  and that .

Therefore, the derivative is

The last step is to substitute  for  in the derivative, which will tell us the slope of the tangent line to  at .

### Example Question #1 : First And Second Derivatives Of Functions

Find the second derivative of the given function:

Explanation:

To find the second derivative, first we need to find the first derivative. To find the first derivative we need to use the quotient rule as follows. So for the given function, we get the first derivative to be

Now we have to take the derivative of the derivative. To do this we need to use the quotient rule as shown below.

Thus, we get

### Example Question #1 : Derivative Rules For Sums, Products, And Quotients

Find the derivative of the function

Explanation:

We find the answer using the quotient rule

and the product rule

and then simplifying.

or . The extra brackets in the denominator are optional.

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