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# Calculus questions on differentiation

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A circle of radius 3 inches is inside a square with 12-inch sides (see figure below). How fast is the area between the circle and square changing if the radius is increasing at 4 inches per minute and the sides are increasing at 2 inches per minute?

Find dy/dx in two way (a) by differentiating implicitly and (b) by explicitly solving for y and then differentiating. Then find the value of dx/dy at the given point using your results from both the
implicit and the explicit differentiation.

x^2+5y^2=45, point: (5, 2)

Find dy/dx using implicit differentiation and then find the slope of the line tangent to the graph of the equation at the given point.
y^2-5xy+x^2+21=0 point: (2, 5)

Find dy/dx in two ways: (a) by using the "usual" differentiation patterns and (b) by using logarithmic differentiation.
y=cos^7⁡〖(2x+5)〗

use logarithmic differentiation to Find dy/dx
y=(cos⁡(x) )^x

https://brainmass.com/math/functional-analysis/calculus-questions-differentiation-614124

## SOLUTION This solution is FREE courtesy of BrainMass!

1. A circle of radius 3 inches is inside a square with 12-inch sides (see figure below). How fast is the area between the circle and square changing if the radius is increasing at 4 inches per minute and the sides are increasing at 2 inches per minute?

Solution:

Let the side of square be x, radius of circle be r and the area between square and circle be A.

Area between the circle and square is changing at the rate of square inches per minutes.

2. Find in two way (a) by differentiating implicitly and (b) by explicitly solving for y and then differentiating. Then find the value of at the given point using your results from both the
implicit and the explicit differentiation.

, point: (5, 2)

Solution:

(a)

Differentiate on each side

Now value of dy/dx at the given point,

(b)

Now value of dy/dx at the given point,
Answers are the same for dx/dy from both the methods.

3. Find using implicit differentiation and then find the slope of the line tangent to the graph of the equation at the given point.
point: (2, 5)

Solution:

Differentiate on each side

To find the slope of the tangent line, put x =2 and y = 5 in dy/dx

Slope is undefined so tangent will be vertical line at the point (2, 5).

4. Find in two ways: (a) by using the "usual" differentiation patterns and (b) by using logarithmic differentiation.

Solution:

(a)

(b)

Take log on each side

5. use logarithmic differentiation to Find

Solution:

Take log on each side