Maths - Chain Rule
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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?
\[2\]