Maths - Chain Rule

Pearson Edexcel Mathematics 2022

Created: February 03, 2021 | Read markdown | About these notes | View in context | Study these flashcards |

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Flashcards

The chain rule and when to use it

What is the chain rule?

chain-rule-formula
\[\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}\]

What new variable do you introduce when using the chain rule?

chain-rule-substitution-variable
\[u\]

When do you apply the chain rule?

chain-rule-when-to-apply

When you have composite functions.

Worked substitution example

\[y = (3x + 4)^5\]

What substitution would you make in order to differentiate?

chain-rule-substitution-choice
\[u = 3x + 4 \\\\ y = u^5\]
\[y = u^5\]

What is $\frac{dy}{du}$?

derivative-u-to-the-fifth
\[5u^4\]
\[u = 3x -1\]

What is $\frac{du}{dx}$?

derivative-linear-inner
\[3\]
\[y = (3x + 4)^5 \\ u = 3x + 4 \\ y = u^5 \\ \frac{dy}{du} = 5u^4 \\ \frac{du}{dx} = 3\]

What is $\frac{dy}{dx}$?

chain-rule-combine-result
\[15(3x + 4)^4\]

The “wiggle” analogy

\[\ln \boxed{beans}\]

How much does the $\ln$ bit “wiggle”?

ln-wiggle-factor
\[\frac{1}{\boxed{beans}}\]
\[\ln \boxed{beans}\]

What’s the derivative?

ln-derivative-wiggle
\[\frac{1}{\boxed{beans}} \times (\frac{dy}{d(beans)} \boxed{beans})\]

What’s an analogy for the chain rule?

chain-rule-wiggle-analogy

Multiplying how much stuff wiggles.

Practice derivatives

\[\ln (2x + 5)\]

What’s the derivative?

derivative-ln-2x-plus-5
\[\frac{2}{2x + 5}\]
\[(3x-1)^4\]

What’s the derivative?

derivative-3x-minus-1-fourth
\[12(3x-1)^3\]
\[(2x + 3)^5\]

What’s the derivative?

derivative-2x-plus-3-fifth
\[10(2x + 3)^4\]
\[(x + 7)^9\]

What’s the derivative?

derivative-x-plus-7-ninth
\[9(x + 7)^8\]
\[\ln \boxed{2x}\]

How much “wiggle” does the $2x$ contribute?

ln-2x-inner-wiggle
\[2\]


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