These are my complete notes for the 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.
# 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 .
(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 .
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 .
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.
#
Rule .
To find the derivative of a logarithm, for a > 0 and a ≠ 1, you must use the chain rule:
$$\frac{d}{dx} \left( \log_a x \right) = \frac{1}{x \ln(a)} \times x'$$
This means that after 1/xln(a), the chain rule would be applied for 2nd value (not the base - in this case a is the base).
Example: (log42x)' = (1 / 2xln(4)) × 2
= 1/xln(4)
It is the ln of the 1st value (the base) of the logarithm that is used.
Example: (log42x)' = (1 / 2xln(4)) × 2
= 1/xln(4)
It is the ln of the 1st value (the base) of the logarithm that is used.
#
Rule .
If you have to find the derivative of a function that requires using the chain rule to multiply by the derivative of a part of the function, and that part of the function is a constant such as ln(2) or log22, e.g. nowhere there is an x, then the derivative equals zero. eln(2), for example.
