Module 12  Rules of Differentiation  
Introduction  Lesson 1  Lesson 2  Self Test  
Lesson 12.2: Rules of Differentiation: Applications and Graphical Support  
A calculatorproduced graph cannot provide confirmation that your analytic work is correct because calculator graphs are sometimes flawed. However, graphs that match can be considered support that your work is probably correct. In this lesson you will use rules to find derivatives of several functions and then use your TI83 to provide support for your results. The Power Rule One of the most basic functions is a power of x. Its derivative can be found by using the Power Rule. The Power Rule says Finding a Tangent Line
The Power Rule can be used to find the derivative of the function y = x^{3}. The derivative can then be used to find the slope and the equation of the line tangent to y = x^{3} at the point (1,1). The easiest way to write the equation is to use the by using the
The derivative of the function y = x^{3} is The derivative evaluated at x = 1 is 3, so the equation of the tangent line is Support this result by graphing y = x^{3} and y = 3(x  1) + 1 in a [3, 3, 1] x [5, 5, 1] viewing window. The line appears to be tangent to the curve at x = 1, which provides graphical support that the equation of the tangent line is correct. Other Derivative Rules Many derivatives are found by using more than one rule. The rules for finding the derivative of a constant multiple of a function and the sum and difference of two functions can be used to find the derivatives of polynomials. The derivative of the polynomial found by using the rules can then be supported graphically. The Derivative of a Constant Function Rule If f is the function with the constant value c, then The Constant Multiple Rule The Constant Multiple Rule says: If f is a differentiable function of x and c is a constant then The Sum and Difference Rule The Sum and Difference Rule says: If f and g are differentiable functions of x, then Derivative of a Polynomial The previous two rules, along with the Power Rule, can be used to find the derivative of any polynomial. For example, the derivative of y = x^{4}  2x^{3} + 5x^{2}  7x + 11 is y' = 4x^{3}  6x^{2} + 10x  7 . Graphs can provide support for the result.
Both graphs appear to be the same, which is graphical support that the derivative defined in Y_{3} is correct. The Product Rule What is the derivative of the product of two differentiable functions? In other words, if u and v are differentiable functions of x then what is the derivative of their product, ? After using the Sum and Difference Rule you might think there is a similar rule for the derivative of a product. However, you can illustrate that this is not the case by displaying a counterexample with your calculator. A Counterexample Show that
The two graphs are not the same, which is graphical evidence that . The Product Rule for derivatives actually says: Using this rule to find the derivative of y = x^{3} sin x gives Provide graphical support for this derivative.
Tracing on both graphs produces the same coordinates. This is graphical support that the derivative defined in Y_{3} is correct. The Quotient Rule The Quotient Rule for derivatives says: If u and v are differentiable functions of x then
12.2.1 Use the Quotient Rule to find the derivative of . Support your result graphically. Click here for the answer. 

< Back  Next >  
©Copyright
2007 All rights reserved. 
Trademarks

Privacy Policy

Link Policy
