The following problems require the use of the chain rule. The chain rule is a rule for differentiating compositions of functions. In the following discussion and solutions the derivative of a function

.

However, we rarely use this formal approach when applying the chain rule to specific problems. Instead, we invoke an intuitive approach. For example, it is sometimes easier to think of the functions *f* and *g* as ``layers'' of a problem. Function *f* is the ``outer layer'' and function *g* is the ``inner layer.'' Thus, the chain rule tells us to first differentiate the outer layer, leaving the inner layer unchanged
(the term *f*'( *g*(*x*) ) ) , then differentiate the inner layer (the term *g*'(*x*) ) . This process will become clearer as you do the problems. In most cases, final answers are given in the most simplified form.

*PROBLEM 1 :*Differentiate .Click HERE to see a detailed solution to problem 1.

*PROBLEM 2 :*Differentiate .Click HERE to see a detailed solution to problem 2.

*PROBLEM 3 :*Differentiate .Click HERE to see a detailed solution to problem 3.

*PROBLEM 4 :*Differentiate .Click HERE to see a detailed solution to problem 4.

*PROBLEM 5 :*Differentiate .Click HERE to see a detailed solution to problem 5.

*PROBLEM 6 :*Differentiate .Click HERE to see a detailed solution to problem 6.

*PROBLEM 7 :*Differentiate .Click HERE to see a detailed solution to problem 7.

*PROBLEM 8 :*Differentiate .Click HERE to see a detailed solution to problem 8.

*PROBLEM 9 :*Differentiate .Click HERE to see a detailed solution to problem 9.

*PROBLEM 10 :*Differentiate .Click HERE to see a detailed solution to problem 10.

*PROBLEM 11 :*Differentiate .Click HERE to see a detailed solution to problem 11.

*PROBLEM 12 :*Differentiate .Click HERE to see a detailed solution to problem 12.

*PROBLEM 13 :*Differentiate .Click HERE to see a detailed solution to problem 13.

*PROBLEM 14 :*Differentiate .Click HERE to see a detailed solution to problem 14.

*PROBLEM 15 :*Differentiate .Click HERE to see a detailed solution to problem 15.

*PROBLEM 16 :*Differentiate .Click HERE to see a detailed solution to problem 16.

*PROBLEM 17 :*Differentiate .Click HERE to see a detailed solution to problem 17.

*PROBLEM 18 :*Differentiate .Click HERE to see a detailed solution to problem 18.

*PROBLEM 19 :*Assume that*h*(*x*) =*f*(*g*(*x*) ) , where both*f*and*g*are differentiable functions. If*g*(-1)=2,*g*'(-1)=3, and*f*'(2)=-4 , what is the value of*h*'(-1) ?Click HERE to see a detailed solution to problem 19.

*PROBLEM 20 :*Assume that , where*f*is a differentiable function. If and , determine an equation of the line tangent to the graph of*h*at*x*=0 .Click HERE to see a detailed solution to problem 20.

*PROBLEM 21 :*Determine a differentiable function*y*=*f*(*x*) which has the properties and .Click HERE to see a detailed solution to problem 21.

Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :

Tue May 6 17:21:40 PDT 1997