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C11 · Unit C3 Composite, implicit and inverse derivatives

Implicit differentiation and higher derivatives

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 3.2, 3.6
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

When x and y satisfy a relation, treat y as a function of x on a suitable local branch. Differentiate every term with respect to x; a y derivative contributes y minutes by chain rule. Solve for y minutes only after collecting terms. A second derivative differentiates y minutes along the curve and may require substitution of the first derivative.

Worked example

From x squared +y squared equals 25, 2x+2yy minutes equals 0 gives y minutes equals minus x over y when y is not equal to 0. Differentiating again yields y seconds equals minus 1 over y+x y minutes /y squared equals minus (x squared +y squared)/y cubed equals minus 25 over y cubed. On the upper semicircle it is negative, matching concavity down.

Explore the mathematics

Implicit differentiation and higher derivatives

Differentiate a circle relationship and locate its local slope.

Setting 3 of 5

Scroll sideways to see the full diagram.

Can you find a tangent without first solving the entire relation for y?xy123x²+y²=7, upper semicircle

Predict before continuing

Can you find a tangent without first solving the entire relation for y?

Read the complete explanation
  1. Can you find a tangent without first solving the entire relation for y?
  2. Differentiate both sides with respect to x: 2x+2y·y′=0.
  3. Thus y′=−x/y where y≠0; the upper curve has a negative slope for x>0.
  4. At (1,√6), y′=−1/√6≈-0.4082. At y=0 this formula cannot describe a finite slope.
See the idea: Implicit differentiation and higher derivatives
See the idea: Implicit differentiation and higher derivativesThe upper semicircle x²+y²=25 is concave down: y″=−25/y³<0 when y>0. The formula y′=−x/y excludes the two endpoints where y=0.xy-50-2.51.5032.54.556

The upper semicircle x²+y²=25 is concave down: y″=−25/y³<0 when y>0. The formula y′=−x/y excludes the two endpoints where y=0.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Fixing y at a point before differentiation incorrectly treats a changing quantity as constant. A denominator zero requires local analysis, not an automatic conclusion in every implicit curve.

Investigate and explain

Find horizontal and vertical tangents of a circle. Use the equation and local geometry where the slope formula is undefined.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.