# Maths - Chain Rule

> Source: https://ollybritton.com/notes/a-level/maths/topics/chain-rule/ · Updated: 2021-02-03 · Tags: maths, further-maths, differentiation

## See Also
- [Maths - Differentiation](https://ollybritton.com/notes/a-level/maths/topics/differentiation/)
- <https://betterexplained.com/articles/derivatives-product-power-chain/>

## Flashcards
##### What is the chain rule??
$$
\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}
$$

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

##### When do you apply the chain rule??
When you have composite functions.

##### $$y = (3x + 4)^5$$ What substitution would you make in order to differentiate??
$$
u = 3x + 4 \\\\
y = u^5
$$

##### $$y = u^5$$ What is $\frac{dy}{du}$??
$$
5u^4
$$

##### $$u = 3x -1$$ What is $\frac{du}{dx}$??
$$
3
$$

##### $$y = (3x + 4)^5 \\ u = 3x + 4 \\ y = u^5 \\ \frac{dy}{du} = 5u^4 \\ \frac{du}{dx} = 3$$ What is $\frac{dy}{dx}$??
$$
15(3x + 4)^4
$$

##### $$\ln \boxed{beans}$$ How much does the $\ln$ bit "wiggle"??
$$
\frac{1}{\boxed{beans}}
$$

##### $$\ln \boxed{beans}$$ What's the derivative??
$$
\frac{1}{\boxed{beans}} \times (\frac{dy}{d(beans)} \boxed{beans})
$$

##### What's an analogy for the chain rule??
Multiplying how much stuff wiggles.

##### $$\ln (2x + 5)$$ What's the derivative??
$$
\frac{2}{2x + 5}
$$

##### $$(3x-1)^4$$ What's the derivative??
$$
12(3x-1)^3
$$

##### $$(2x + 3)^5$$ What's the derivative??
$$
10(2x + 3)^4
$$

##### $$(x + 7)^9$$ What's the derivative??
$$
9(x + 7)^8
$$

##### $$\ln \boxed{2x}$$ How much "wiggle" does the $2x$ contribute??
$$
2
$$

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