The chain rule - dy/du notation Revision Card | Tope Teaches Maths

CalculusAdditional / Further Maths

Differentiate a composite function using and .

Choose the inner expression and write as a function of .
Use .
Replace by its expression in in the final answer.

Example 1

Differentiate with respect to .

Choose the inner expression.

Express the outer function in terms of

.

Differentiate the inside with respect to

.

Differentiate the outside with respect to

.

Multiply the two derivatives.

Replace

by the original inside expression.

Example 2

Differentiate with respect to .

Choose the inner expression.

Express the outer function in terms of

.

Differentiate the inside with respect to

.

Differentiate the outside with respect to

.

Multiply the two derivatives.

Replace

by the original inside expression.

Your turn

Question 1

Differentiate with respect to .

Check answer

Choose the inner expression.

Express the outer function in terms of

.

Differentiate the inside with respect to

.

Differentiate the outside with respect to

.

Multiply the two derivatives.

Replace

by the original inside expression.

Question 2

Differentiate with respect to .

Check answer

Choose the inner expression.

Express the outer function in terms of

.

Differentiate the inside with respect to

.

Differentiate the outside with respect to

.

Multiply the two derivatives.

Replace

by the original inside expression.

Question 3

Differentiate with respect to .

Check answer

Choose the inner expression.

Express the outer function in terms of

.

Differentiate the inside with respect to

.

Differentiate the outside with respect to

.

Multiply the two derivatives.

Replace

by the original inside expression.

Question 4

Differentiate with respect to .

Check answer

Choose the inner expression.

Express the outer function in terms of

.

Differentiate the inside with respect to

.

Differentiate the outside with respect to

.

Multiply the two derivatives.

Replace

by the original inside expression.

Question 5

Differentiate with respect to .

Check answer

Choose the inner expression.

Express the outer function in terms of

.

Differentiate the inside with respect to

.

Differentiate the outside with respect to

.

Multiply the two derivatives.

Replace

by the original inside expression.

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