Maths - Chain Rule
See Also
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.
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\[y = (3x + 4)^5\]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.
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\[\ln (2x + 5)\]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??
\[2\]