# 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 #61 : Computation Of Derivatives

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 #62 : Computation Of Derivatives

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 #63 : Computation Of Derivatives

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 #64 : Computation Of Derivatives

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 #26 : 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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