More Calculus |
| Vol. 3, Why Slopes & More Maths, also gives starter lessons for differential and integral calculus. |
Chain Rule for Differentiable Functions
the general case for real-valued functions of a single real variable
The foregoing exercises in deriving the power rule, the chain for linear functions, the chain rule for powers and the chain rule for polynomials provides, we hope the background and experience needed to understand the statement of the chain rule for differentiable functions in general.
The polynomial outermost function special case of the chain rule was explained in earlier lessons. That case should help you understand the statement of the chain rule for the general case where the outer function f need not be polynomial (think of trig and exponential and logarithmic functions)
Formal Statement:
Theorem (Chain Rule): Suppose z = f (y) is differentiable at y = b, so that f '(b) is a real number. Suppose y = g(x) is differentiable at x = a with g(a) = b. Then z = f(g(x)) = h(x) is differentiable at x = a with
h ' (a) = f '(b) g'(a)
Note the conditions on the functions f and g. If you are interested in proofs, or in skill and concept perfection, you should remember the conditions. In practice, these condition are satisfied at most points. Learn to do first and worry about the conditions later
An informal statement of the chain rule
