Chain Rule

    With a lot of work, we can find derivatives without using the chain rule either by expanding a polynomial, by using another differentation rule, or maybe by using a trigonometric identity. The derivative would be the same in either approach; however, the chain rule allows us to find derivatives that would otherwise be very difficult to handle. This topic gives plenty of examples of the chain rule as well an easily understandable proof of the chain rule.  

Proposition (Chain Rule) If the derivatives and both exist, and is the composite function defined by then exists and is given by the product



In Leibniz notation, if and are both differentiable functions, then

Example (Chain Rule) Apply the chain rule.

(a) Let be a function for which If what is Also, if what is

    Solution. By the chain rule



Also, by the chain rule



(b) Let be a function for which and If what is

    Solution. Using the chain rule and the product rule we determine,





Therefore,




(c) Assuming that the following derivatives exists, find



    Solution. Using the chain rule,
    


which is the second derivative evaluated at the function multiplied by the first derivative; while,



is the first derivative evaluated at the first derivative multiplied by the second derivative. When will these derivatives be the same?

(d) Show that if a particle moves along a straight line with position and velocity then its acceleration satisfies Use this formula to find in the case where

    Solution. By the chain rule,



In the case where we determine,



Thus, What does this rate of change represent?

    In using the chain rule we work from the outside to the inside. Basically, the chain rule says that we differentiate the outer function [at the inner function] and then we multiply by the derivative of the inner function.

Proposition (Proof of the Chain Rule) Suppose is a differentiable function of which is a differentiable function of Then is a differentiable function of and



    Proof. We wish to show and will do so by using the definition of the derivative for the function with respect to namely,



To better work with this limit let's define an auxiliary function:



Let then three properties of the function are

    (i) provided
    
    (ii)
     
    (iii) is continuous at since

Now we can rewrite as follows:

Example (Using the Chain Rule) Use the chain rule and show all steps.

(a) Find an equation of the tangent line to the graph of the function at the point

    Solution. By using the chain rule we determine,



and so Therefore, an equation of the tangent line is which implies to   Here is a plot of both and the tangent line to at



(b) Determine the point(s) at which the graph of has a horizontal tangent.

    Solution. By using the chain rule we determine,



which simplifies to Thus the only point where has a horizontal tangent line is and these can be seen in the following sketch.

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