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CalculusBy Educator Today

C10 · Unit C3 Composite, implicit and inverse derivatives

Chain rule and choosing a differentiation method

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 3.1, 3.5
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

A composite function changes through its inner input. The chain rule gives d f(g(x))/dx equals f minutes (g(x))g minutes (x). Work from outer operation to inner operation and retain parentheses. Products of compositions may require both product and chain rules. Method selection begins by identifying the expression’s structure, not by looking for one familiar symbol.

Worked example

For y equals sin(x squared +1), outer derivative is cosine and inner derivative is 2x, so y minutes equals 2x cos(x squared +1). For y equals (3x minus 2) to the power 4, y minutes equals 4(3x minus 2) cubed times 3 equals 12(3x minus 2) cubed.

Explore the mathematics

Chain rule and choosing a differentiation method

Separate an inner and an outer function, then multiply their rates.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict how an inner change becomes an outer change.xy12345y=(x+1)²; inner u=x+1, outer y=u²

Predict before continuing

Predict how an inner change becomes an outer change.

Read the complete explanation
  1. Predict how an inner change becomes an outer change.
  2. The outer rate is dy/du=2u; the inner rate is du/dx=1.
  3. Multiply linked rates: dy/dx=(dy/du)(du/dx)=2(x+1).
  4. At x=4, the derivative is 10. For an inner ax+b, the extra factor a is essential.
Chain rule and choosing a differentiation method · visual worked method
Chain rule and choosing a differentiation method · visual worked methodFor y=sin(x²+1), outer derivative is cosine and inner derivative is 2x, so y′=2x cos(x²+1). For y=(3x−2)⁴, y′=4(3x−2)³·3=12(3x−2)³.Worked example · 1/2For y=sin(x²+1), outer derivativeis cosine and inner derivative is2x, so y′=2x cos(x²+1). Fory=(3x−2)⁴,Worked example · 2/2y′=4(3x−2)³·3=12(3x−2)³.

For y=sin(x²+1), outer derivative is cosine and inner derivative is 2x, so y′=2x cos(x²+1). For y=(3x−2)⁴, y′=4(3x−2)³·3=12(3x−2)³.

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

A common misconception

Forgetting the inner derivative changes both scale and sometimes sign. The outer function must still be evaluated at the original inner expression.

Investigate and explain

Create a dependency tree for sin((x squared +1) cubed). Label the derivative factor contributed by each layer.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.