# AP Calculus BC : Chain Rule and Implicit Differentiation

## Example Questions

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### Example Question #21 : Chain Rule And Implicit Differentiation

Find :

, where  is a constant.

Explanation:

The derivative of the function is equal to

and was found using the following rules:

The constant may seem intimidating, but we treat it as another constant!

### Example Question #21 : Chain Rule And Implicit Differentiation

Find  from the following equation:

, where  is a function of x.

Explanation:

To find the derivative of  with respect to x, we must differentiate both sides of the equation with respect to x:

The derivatives were found using the following rules:

Solving for , we get

Note that the chain rule was used because of the exponential and because  is a function of x.

### Example Question #22 : Chain Rule And Implicit Differentiation

Find the first derivative of the following function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

Note that the chain rule was used on the secant function as well as the natural logarithm function.

### Example Question #23 : Chain Rule And Implicit Differentiation

Find :

Explanation:

To determine , we must take the derivative of both sides of the equation with respect to x:

The derivatives were found using the following rules:

Rearranging and solving for , we get

### Example Question #21 : Chain Rule And Implicit Differentiation

A curve in the xy plane is given implictly by

.

Calculate the slope of the line tangent to the curve at the point .

Explanation:

Differentiate both sides with respect to  using the chain rule and the product rule as:

Then solve for  as if it were our unknown:

.

Finally, evaluate  at the point  to obtain the slope through that point:

.

### Example Question #22 : Chain Rule And Implicit Differentiation

squircle is a curve in the xy plane that appears like a rounded square, but whose points satisfy the following equation (analogous to the Pythagorean theorem for a circle)

where the constant  is the "radius" of the squircle.

Using implicit differentiation, obtain an expression for  as a function of both  and .

Explanation:

Differentiate both sides of the equation with respect to , using the chain rule on the  term:

Then solve for  as if it were our unknown:

.

Comparing this to the figure, our answer makes sense, because the slope of the squircle is  wherever  (as it crosses the  axis) and undefined (vertical) wherever  (as it crosses the  axis). Lastly, we note that in the first quadrant (where  and ), the slope of the squircle is negative, which is exactly what we observe in the figure.

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