CalculusCourse skill

The chain rule - notation

Differentiate a composite function using and .

  • Write , with the inner function and the outer function.
  • Use .
  • Evaluate at , then multiply by the inner derivative.

Example 1

Differentiate with respect to .

Choose the inner function.Choose the outer function, so ().Differentiate the inner function.Differentiate the outer function.Evaluate the outer derivative at the whole inner function.Multiply by the inner derivative.

Example 2

Differentiate with respect to .

Choose the inner function.Choose the outer function, so ().Differentiate the inner function.Differentiate the outer function.Evaluate the outer derivative at the whole inner function.Multiply by the inner derivative.

Your turn

Question 1

Differentiate with respect to .

Check answer
Choose the inner function.Choose the outer function, so ().Differentiate the inner function.Differentiate the outer function.Evaluate the outer derivative at the whole inner function.Multiply by the inner derivative.
Question 2

Differentiate with respect to .

Check answer
Choose the inner function.Choose the outer function, so ().Differentiate the inner function.Differentiate the outer function.Evaluate the outer derivative at the whole inner function.Multiply by the inner derivative.
Question 3

Differentiate with respect to .

Check answer
Choose the inner function.Choose the outer function, so ().Differentiate the inner function.Differentiate the outer function.Evaluate the outer derivative at the whole inner function.Multiply by the inner derivative.
Question 4

Differentiate with respect to .

Check answer
Choose the inner function.Choose the outer function, so ().Differentiate the inner function.Differentiate the outer function.Evaluate the outer derivative at the whole inner function.Multiply by the inner derivative.
Question 5

Differentiate with respect to .

Check answer
Choose the inner function.Choose the outer function, so ().Differentiate the inner function.Differentiate the outer function.Evaluate the outer derivative at the whole inner function.Multiply by the inner derivative.
← All Calculus cardsReport a mistake

Related cards

The chain rule - dy/du notationDifferentiating natural logarithmsThe product rule - u, v notation