These are my complete notes for Chain Rule in Differential Calculus.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Summary of Chain Rule (Differential Calculus)
?. Chain Rule.
(f(g(x)))' = f'(g(x)) × g'(x)
# Integers with Exponential Powers:
For a > 0, and a ≠ 1, (au)' = au × ln(a) × (du/dx)
Without the chain rule for the exponent, this simplifies to (ax)' = ax × ln(a).
# Rule 159. (ex)1 is just ex. Whenever finding the derivative of an equation involving ex, just copy that original ex expression and multiply by the chain rule.
# Rule 160. For other expressions where x is part of the exponent, like 9-x or 3csc(x), you copy the original equation, multiply by ln of the base, and by the derivative of the exponent: 3csc(x) = ln(3) × 3csc(x) × -csc(x)cot(x). See the Integers with Exponential Powers blue section.
# Rule 161. For finding the derivative of an equation with ln, know that (ln(x))' = 1/x, with the x changing as needed. The derivative is just the changed value of 1/x multiplied by the derivative of the x-value: (ln(x²))' = (1/x²) × 2x.
