Chapter 3 Differentiation Rules

Transcription

1 Chapter 3 Differentiation Rules Derivative constant function if c is any real number, then Example: The Power Rule: If n is a positive integer, then Example: Extended Power Rule: If r is any real number, then Example: * + Example: * + Example: * + Example: Find equations of the tangent line and normal line to the curve at the point. Illustrate by graphing the curve and these lines. * + [ ] * + So the slope of the tangent line at (1, 1) is. Therefore an equation of the tangent line is or The normal line is perpendicular to the tangent line, so its slope is the negative reciprocal of, that is,. Thus an equation of the normal line is or Calculus _Math 201 Page 1

2 Constant Multiple Rule: If f is differentiable at x and c is any real number, then Example: Example: Example: * + Sum and Difference Rules If and are differentiable at, then Example: Example: Example: * + * + [ ] Example: Find if Example: If, then Calculus _Math 201 Page 2

3 Definition of the Number e e is the number such that h e 1 lim 1 h 0 h Derivative of the Natural Exponential Function Example: At what point on the curve is the tangent line parallel to the line? Solution Since, we have. Let the x-coordinate of the point in question be a. Then the slope of the tangent line at that point is. This tangent line will be parallel to the line if it has the same slope, that is, 2. Equating slopes, we get Therefore the required point is. The Product Rule If and are differentiable at x, then so is the product, and Example: a) if, find b) Find the nth derivative, Solution. a) b) Using the Product Rule a second time, we get Further applications of the Product Rule give In fact, each successive differentiation adds another term Example: Differentiate the function Solution. Using the Product Rule, we have, so ( ) ( ) Calculus _Math 201 Page 3

4 The Quotient Rule If f and g are both differentiable at x and if is differentiable at x and, then Example: Find for Example: Find an equation of the tangent line to the curve at the point ( ). According to the Quotient Rule, we have * + So the slope of the tangent line at ( ) is This means that the tangent line at ( ) is horizontal and its equation is. [See Figure] TABLE OF DIFFERENTIATION FORMULAS ( ) Calculus _Math 201 Page 4

5 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS Example: Find if. Example: Differentiate a horizontal tangent?. For what values of x does the graph of. have Since is never 0, we see that when, and this occurs when, where n is an integer Example: Find the 27 th derivative of. The first few derivatives, We see that the successive derivatives occur in a cycle of length 4 and, in particular, whenever is a multiple of 4. Therefore and, differentiating three more times, we have Example: Calculate lim xcot x x 0. Here we divide numerator and denominator by x: x cos x cos x lim cos x x 0 cos 0 lim xcot x lim lim 1 x 0 x 0 sin x x 0 sin x sin x lim 1 x x 0 x Calculus _Math 201 Page 5

6 The Chain Rule If g is differentiable at x and f is differentiable at g(x), then the composition defined by ( ) is differentiable at x and is given by the product. In Leibniz notation, if and then Example: Find if Solution 1: where and We have Solution 2: If we let then Example: Differentiate a) b) Solution :a) in Let and as. Solution :b) and as. Let The Power Rule Combined With The Chain Rule If is any real number and is differentiable, then Alternatively, Calculus _Math 201 Page 6

7 Example: Differentiate if Let And Example: Find if Then Example: Differentiate Noticing that each term has the common factor, we could factor it out and write the answer as Example: Differentiate Derivatives of Exponential Functions Generalized derivative formulas Example: Find Calculus _Math 201 Page 7

8 Implicit Differentiation Example: Find Example: a) if Find b) Find an equation of the tangent to the circle at the point a) b) At the point we have and, so An equation of the tangent to the circle at is therefore Example: Use implicit differentiation to find if Example: a) Find if. b) Find the tangent to the folium of Descartes a t the point. c) At what points in the first quadrant is the tangent line horizontal? Calculus _Math 201 Page 8

9 ( ) b) When, So an equation of the tangent to the folium is c)the tangent line is horizontal if. Using the expression for from part (a), we see that when (provided that. Substituting in the equation of the curve, we get * + * + which simplifies to. Since in the first quadrant, we have. If, then. Thus the tangent is horizontal at (0, 0) and at ( ), which is approximately Example: Find if. Example: Find if. ( ) Calculus _Math 201 Page 9

10 Derivatives of Inverse Trigonometric Functions Generalized derivative formulas Example: Find Derivatives of Logarithmic Functions Generalized derivative formulas Calculus _Math 201 Page 10

11 Example: Find let Example: Find let Example: Find if Example: Differentiate Example: Find { if { Example: Differentiate Calculus _Math 201 Page 11

12 Example: Differentiate take ln [ ] Calculus _Math 201 Page 12