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CalculusBy Educator Today

C57 · Unit C3 Composite, implicit and inverse derivatives

Method selection and explaining a strategy

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 3.5; 6.14 BC integration extension
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Before manipulating an expression, name its outer structure: sum, product, quotient, composition or implicit relation. Choose a rule based on that structure. Integration is less automatic: simplify algebra first, look for an inner derivative, and in BC consider parts or partial fractions. A correct final expression is stronger when the chosen method and its domain are explained.

Worked example

For f(x) equals x ln(1+x squared), product rule comes first and chain rule appears inside: f minutes equals ln(1+x squared)+2x squared /(1+x squared). For ∫2x/(1+x squared)dx, substitution directly gives ln(1+x squared)+C. The similar symbols conceal different tasks.

Explore the mathematics

Method selection and explaining a strategy

Change a variable and its differential and limits together.

Setting 3 of 5

Scroll sideways to see the full diagram.

Look for an inside expression and its derivative.xy1Marked input 0.5; displayed curve height 1.25∫₀¹ 2x(x²+1) dx

Predict before continuing

Look for an inside expression and its derivative.

Read the complete explanation
  1. Look for an inside expression and its derivative.
  2. Set u=x²+1, so du=2x dx. The limits become u=1 and u=2.
  3. Integrate ∫₁² u du=[u²/2]₁².
  4. The result is 3/2. If you change variables, transform the limits or substitute back before using the original limits.
Method selection and explaining a strategy · visual worked method
Method selection and explaining a strategy · visual worked methodFor f(x)=x ln(1+x²), product rule comes first and chain rule appears inside: f′=ln(1+x²)+2x²/(1+x²). For ∫2x/(1+x²)dx, substitution directly gives ln(1+x²)+C. The similar symbols conceal different tasks.Worked example · 1/2For f(x)=x ln(1+x²), product rulecomes first and chain rule appearsinside: f′=ln(1+x²)+2x²/(1+x²).For ∫2x/(1+x²)dx,Worked example · 2/2substitution directly givesln(1+x²)+C. The similar symbolsconceal different tasks.

For f(x)=x ln(1+x²), product rule comes first and chain rule appears inside: f′=ln(1+x²)+2x²/(1+x²). For ∫2x/(1+x²)dx, substitution directly gives ln(1+x²)+C. The similar symbols conceal different tasks.

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

A common misconception

Differentiation and integration are inverse in principle, but a derivative rule is not automatically an integration rule with every symbol reversed.

Investigate and explain

Write a one-sentence strategy for five mixed problems before solving. Compare it with the method you actually used.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.