Using the Chain Rule
Definition of Composite Functions
A composite function is a function that is made up of another function and comes in the form of f(g(x)). The functions we’re used to typically come in the form of just f(x). For example, a “normal” function would be ln(x), but a composite would be ln(x^2). Here we can see that inside the parentheses of the natural log function, there’s another function: x^2.
When dealing with these types of functions, we call f(x) the “outer” function and g(x) the “inner” function. For example, in the example from before, ln(x) would be our outer function and x2 would be our inner one. Identifying which function is outer/inner is super important in using the chain rule!
Here’s some practice with getting used to determining f and g!
Chain Rule Method
The chain rule method is used when taking the derivative of composite functions! In mathematical terms, the chain rule method is the following:
This is said out loud as “The derivative of f(g(x)) is equal to the derivative of f(g(x)) multiplied by the derivative of g(x)”
The chain rule can use several other methods of taking derivatives such as the power rule
, quotient rule
, trigonometric identities, and others
. Making sure you’re confident in these other methods is your key to success with the chain rule😎
The biggest parts of using the chain rule is (1) identifying when to use it, (2) identifying f(g(x)) and g(x), and (3) applying the method. Steps (1) and (2) simply require identifying if there’s a composite function in what you’re taking the derivative of and, if so, determining the inner and outer functions (as explained above).
Step (3) can get a bit tricky! Essentially, when taking the derivative of f(g(x)), you’ll be treating the function g(x) as a single variable. For example, if we had the following function:
Common Pitfalls + Helpful Tips!
In trigonometric or natural log functions being raised to the nth power (ex. ln3(x) or sin2(2x) ), oftentimes the exponent can be confusing to work with. Rewriting the function by putting the exponent outside of a pair of parentheses can help with this! (ex. ln3(x) → (ln(x))3 )
f’(g(x)) should be multiplied by g’(x), not added!
Be sure to include ALL of g(x), constants and all, when substituting it back for ‘x’ after taking the derivative of f(x).
Along with tip (3), make sure that what you’re substituting for ‘x’ is g(x) NOT g’(x)
The chain rule isn’t limited to just f(g(x))! We can also have f(g(h(x))) and f(g(h(i(x)))) and so on and so forth. Although it may seem daunting, the process of taking the derivative for these functions is the exact same as before (check out the next section for some examples 👇) As always, make sure to analyze all parts of the function given!
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