Basic Rules of Differentiation

1. Basic Rules of Differentiation By Dr. Julia Arnold and Ms. Karen Overman using Tan’s 5th edition Applied Calculus for the managerial , life, and social sciences

2. Basic Rules of Differentiation Using the definition of the derivative, can be tiresome. In this lesson we are going to learn and use some basic rules of differentiation that are derived from the definition. For these rules, let’s assume that we are discussing differentiable functions.

3. Rule 1: Derivative of a constant , for any constant,c. This rule states that the derivative of a constant is zero. For example,

4. Rule 2:The Power Rule , where n is any real number This rule states that the derivative of x raised to a power is the power times x raised to a power one less or n-1. For example, Notice that the derivative is the original power, 5 times x raised to the fourth, which is one less than 5.

5. Rule 3: Derivative of a Constant Multiple of a Function , where c is a constant This rule states that the derivative of a constant times a function is the constant times the derivative of the function. For example, find the derivative of

6. Rule 4: Derivative of a Sum or Difference This rule states that the derivative of a sum or difference is the sum or difference of the derivatives. For example, find the derivative of x2 +2x -3 The derivative of x squared is done by the Power Rule (2), the derivative of 2x is done by rule 3 and power rule and the derivative of 3, a constant is 0.

7. More Examples: Find You’ll notice none of the basic rules specifically mention radicals, so you should convert the radical to its exponential form, X1/2 and then use the power rule.

8. More Examples: Find Again, you need to rewrite the expression so that you can use one of the basic rules for differentiation. If we rewrite the fraction as x-2 ,then we can use the power rule.

9. More Examples: Find Rewrite the expression so that you can use the basic rules of differentiation. Now differentiate using the basic rules.

10. Another example: Find the slope and equation of the tangent line to the curve at the point (1,3). Recall from the previous chapter, the derivative gives the slope of the tangent to the curve. So we will need to find the derivative and evaluate it at x = 1 to find the slope at the point (1,3). Then we’ll use the slope and the point to write the equation of the tangent line using the point slope form.

11. Find the slope and equation of the tangent line to the curve at the point (1,3). First find the derivative of the function. Now, evaluate the derivative at x = 1 to find the slope at (1,3).

12. Example continued: Now we have the slope of the tangent line at the point (1,3) and we can write the equation of the line. Recall to write the equation of a line, start with the point slope form and use the slope, 4 and the point (1,3).

13. The graph below shows the curve in blue and the tangent line at the point (1,3), in red.

14. Another example: Find all the x values where has a horizontal tangent line. Find the derivative. Since horizontal lines have a slope of 0, set the derivative equal to 0 and solve for x. Thus the x values where the function has horizontal tangents is at x = -1 , -1/3.

15. The graph below shows the function from the last example and the horizontal tangent lines at x=-1 and x=-1/3.