Flashcards in 6.2.2 Applying Implicit Differentiation Deck (9):
Applying Implicit Differentiation
• 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.
- To find the derivative of this relation you must use
- 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.
What is the equation of the line tangent to the curve xy = 4 at the point (2, 2)?
( y − 2) = −1 (x − 2)
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?
Suppose a curve is defined by the equation y/y−x=x^2−1.Find dx/dy.
What is the equation of the line tangent to the curve
x^ 2 + y^ 2 = 25 at the point (0, −5)?
y = −5
Suppose a curve is defined by the equation x^2−xy+y^3=8.Find dx/dy.
What is the equation of the line tangent to the curve x^ 2 + y^ 2 = 100 at the point (6, 8)?
y − 8 = −3/4 (x − 6)