# AP Calculus AB : Chain rule and implicit differentiation

## Example Questions

### Example Question #71 : Chain Rule And Implicit Differentiation

Find the derivative using the chain rule.

Explanation:

Use the chain rule to find the derivative.

### Example Question #72 : Chain Rule And Implicit Differentiation

Find the derivative using the chain rule.

Explanation:

Use the chain rule to find the derivative.

### Example Question #73 : Chain Rule And Implicit Differentiation

Find the derivative using the chain rule.

Explanation:

Use the chain rule to find the derivative.

### Example Question #74 : Chain Rule And Implicit Differentiation

.

Which of the following expressions is equal to  ?

Explanation:

Differentiate both sides with respect to :

By the sum rule:

By the chain rule:

Applying some algebra:

### Example Question #75 : Chain Rule And Implicit Differentiation

Which of the following is equal to  ?

Explanation:

Differentiate both sides with respect to :

Apply the sum, difference, and constant multiple rules:

In the first term, apply the chain rule; in the second, apply the constant multiple rule:

Apply the power rule:

Now apply some algebra:

### Example Question #71 : Chain Rule And Implicit Differentiation

We have three functions,

Find the derivative of

Given that

Explanation:

So now this is a three layer chain rule differentiation. The more functions combine to form the composite function the harder it will be to keep track of the derivative. I find it helpful to lay out each equation and each derivative, so:

Then a three layer chain rule is just the same as a two layer, except... there's one more layer!

It is still the outermost layer evaluated at the inner layers, and then move another layer in and repeat

### Example Question #77 : Chain Rule And Implicit Differentiation

Find  at  with the equation

Explanation:

So with implicit differentiation, you are going to be taking the derivative of every variable, in the entire equation. Every time you take the derivative of a variable, you have it's rate of change multiplied on the right. In this case, dx or dy.

The result of the derivative is:

The first step is to create the  term that you are looking to solve for. This is done by dividing the entire equation by  to get it on the bottom of the fraction. After distributing this division to each term, the dx in the first term will cancel with itself, and you will be left with one term that is multiplied by . At that point, you want to get the term with  onto its own side. This can be accomplished by subtracting the  to the right side of the equation.

The result so far:

Then to finish getting  on its own, you divide  to the right side, ending up with:

So now looking at the question, we know that , so in order to figure out  we need to plug  into the equation.

This gives us .

So this gives us two possible answers:

### Example Question #72 : Chain Rule And Implicit Differentiation

Find the derivative of

Explanation:

So the derivative of a natural log  is always equal to  or one over whatever is inside the natural log. In this case  is inside the natural log, so the derivative of  should be:

But since the inside of the natural log is a function as well, this is the chain rule and the derivative of the natural log will be multiplied by the derivative of the inside, in this case , which is

So the final derivative is

### Example Question #73 : Chain Rule And Implicit Differentiation

Compute the derivative of the following expression:

Explanation:

This problem involves using product rule, and chain rule.

By product rule,

Then, using power rule, and ten chain.

Another application of chain rule to get at the angle,

Taking the derivative of the angle and then simplifying, we get

### Example Question #74 : Chain Rule And Implicit Differentiation

Find the first derivative of the function: