Your comments and suggestions are welcome. PROBLEM 21 : Determine a differentiable function y = f( x) which has the propertiesĬlick HERE to see a detailed solution to problem 21.Ĭlick HERE to return to the original list of various types of calculus problems.Ĭlick HERE to see a detailed solution to problem 20. If and, determine an equation of the line tangent to the graph of h at x=0. PROBLEM 20 : Assume that, where f is a differentiable function.If g(-1)=2, g'(-1)=3, and f'(2)=-4, what is the value of h'(-1) ?Ĭlick HERE to see a detailed solution to problem 19. PROBLEM 19 : Assume that h( x) = f( g( x) ), where both f and g are differentiable functions.

The following three problems require a more formal use of the chain rule. The following seven problems require more than one application of the chain rule.Ĭlick HERE to see a detailed solution to problem 12.Ĭlick HERE to see a detailed solution to problem 13.Ĭlick HERE to see a detailed solution to problem 14.Ĭlick HERE to see a detailed solution to problem 15.Ĭlick HERE to see a detailed solution to problem 16.Ĭlick HERE to see a detailed solution to problem 17.Ĭlick HERE to see a detailed solution to problem 18.

In most cases, final answers are given in the most simplified form.Ĭlick HERE to see a detailed solution to problem 1.Ĭlick HERE to see a detailed solution to problem 2.Ĭlick HERE to see a detailed solution to problem 3.Ĭlick HERE to see a detailed solution to problem 4.Ĭlick HERE to see a detailed solution to problem 5.Ĭlick HERE to see a detailed solution to problem 6.Ĭlick HERE to see a detailed solution to problem 7.Ĭlick HERE to see a detailed solution to problem 8.Ĭlick HERE to see a detailed solution to problem 9.Ĭlick HERE to see a detailed solution to problem 10.Ĭlick HERE to see a detailed solution to problem 11. This process will become clearer as you do the problems. (the term f'( g( x) ) ), then differentiate the inner layer (the term g'( x) ). 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 For example, it is sometimes easier to think of the functions f and g as ``layers'' of a problem. Instead, we invoke an intuitive approach. However, we rarely use this formal approach when applying the chain rule to specific problems. In the following discussion and solutions the derivative of a function h( x) will be denoted by or h'( x). Substitute the value of r 165 m to the obtained equation. Substitute and solve for the growing rate of the oil spot. dA/dt d/dt (r 2 )2r (dr/dt) It is given dr/dt 1.2 meters/minute. The chain rule is a rule for differentiating compositions of functions. Differentiate both sides of the area equation using the chain rule. The following problems require the use of the chain rule.

Word problems calculus chain rule