3.6 Differentials Definition: The Leibniz notation for the derivative, dy/dx can be viewed as a quotient. Definition: The differential of a function f, usually denoted as that is Note: Here, dx is another variable! Thus, the differential is a function of two variables: x and dx. dy Dy P dx=Dx x x+Dx 1

Meaning of the differential: since the derivative plays the role of an instantaneous rate of change, dy is the amount of change if the rate stays the same. The differential is an approximation of the actual amount of change Dy. In fact, this approximation is the simplest, linear: the function is approximated by its tangent (compare with the idea of Newton method, section 6.11). dy Dy P dx=Dx x x+Dx 2

Example: According to the statistics of the past 5 years, the real estate prices in Indianapolis grow according to the dependence where, a is the price 5 years ago, and t is time in years. Use the differential to estimate the growth of the price of your house over the next year if a=300,000. Solution: The differential itself approximate the change of the price. Taking the derivative we find that the price growth starting at the year t and over the interval of the next dt years is Two variables! For the increase over the next year, we take t=5 and dt=1: dp=150,000. 3

Example: Find the differential of the function Solution: Find Substitute that into the definition of the derivative: dy=____________dx 4

6.2 Derivatives of Sine and Cosine Functions Note: Section 6.1, Review of Trigonometry, is to be done at home! Now we continue developing differentiation rules, but the functions now are no longer algebraic. Sine Rule: Proof: Step 1: Step 2: Step 3: Step 4: 5

Proof (cntd): See the book (p.214) for the proof that We apply these limits and write 6

Cosine Rule: Proof: We reduce this case to the previous: Hence, By the chain rule, we can obtain the following generalized rules: 7

Exercises: Differentiate 8

Homework Section 3.6: 1,3,5,7. Section 6.2: 3,7,9,11,17,21,23,25,29,33, 35,37,39,41,45,47. 9