Chain Rule by a concise method
In using the chain rule by the decomposition method we substitute the inner function into the derivative of the outer one at the end. However with a bit of practice, one usually makes the substitution right after each differentiation. That is,
Remember that the Chain Rule can be used on any composite function. A composite function can be thought of as a chain of simpler one operation functions. This function is fghk(x). Starting with x as the clasp, the operations (links) are added one by one, starting with the innermost function, k, and ending with the outermost function, f.
To differentiate the composite function we differentiate the simpler functions (links) one by one, starting with the outermost function, f. Each differentiated piece of chain is multiplied to get the derivative of the full composite function.
This means that you can compute the derivative of a composite function quickly in your head, so long as you can deduce the order of the functions involved in the composition.
Find the derivative of y = sin(ln(5x2 − 2x)).
We need to differentiate the functions one by one from outermost to innermost. Initially we write down the steps required to make the function, in reverse order, until we get back to an expression that can be differentiated by the basic rules. We can then differentiate:
We usually think about what the order of functions is and then deal with them in reverse order.
For derivatives where the Chain Rule is applied once, you should now be practicing simply writing down the complete derivative and tidying the expression.
Differentiate y = sin(2x). This is a composite function. Sine is the outer function, and double is the inner function. Hence = (cos(2x))(2) = 2cos(2x).
Use the concise method to obtain the derivative of each expression:
The next exercises require the application of the chain rule twice. Use the concise method to solve them.
Find the value of f '(x) for each of these functions: