6.2.2 Applying Implicit Differentiation

Question 1

Applying Implicit Differentiation

Answer 1

• Find the derivative of a relation by differentiating each side of its equation implicitly and solving for the derivative as an unknown. This process is called implicit differentiation.
• To find the equation of a line tangent to a curve, you need a point on the line and the slope of the line. To find the slope of the line, you may need to substitute both the x-value and the y-value of the point into the derivative.

Question 2

note

Answer 2

• To find the derivative of this relation you must use
  implicit differentiation.
• Take the derivative of both sides of the equation.
• Notice that you must use the chain rule and the product rule to find the derivative.
• Now substitute the point of tangency into the derivative to find the slope. Notice that you must substitute both the x-value and the y-value.
• Use the point-slope formula to find the equation of the tangent line.
• This implicit equation will require several different
  differentiation rules to differentiate.
• It is a good idea to differentiate each term as a side-problem first and then to combine all of the results at the end of the problem.

Question 3

What is the equation of the line tangent to the curve xy = 4 at the point (2, 2)?

Answer 3

( y − 2) = −1 (x − 2)

Question 4

Suppose a curve is defined by the equation x^2/3+ y^2/3=4.Which set represents all the points on the curve where the line tangent to the curve has slope m=−1?

Answer 4

{(√8,√8),(−√8,−√8)}

Question 5

Suppose a curve is defined by the equation y/y−x=x^2−1.Find dx/dy.

Answer 5

dx/dy=x/y−2x(y−x)^2

Question 6

What is the equation of the line tangent to the curve

x^ 2 + y^ 2 = 25 at the point (0, −5)?

Answer 6

y = −5

Question 7

Suppose a curve is defined by the equation x^2−xy+y^3=8.Find dx/dy.

Answer 7

dx/dy=x−3y^2/2x−y

Question 8

What is the equation of the line tangent to the curve x^ 2 + y^ 2 = 100 at the point (6, 8)?

Answer 8

y − 8 = −3/4 (x − 6)

Question 9

Suppose a curve is defined by the equation
(x+1)^2/25+(y−1)^2/16=1.Which set represents all the points on the curve where the line tangent to the curve has slope m=1?

Answer 9

{(−√41+25/√41,√41−16/√41) , (−√41−25/√41,√41+16/√41)}