2.8, Sections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6: Review Theory
Instructor: Ms. Hoa Nguyen ([e-mail])
2.8 The graphs of a function f (x) and its derivative f (x): f (x) > 0 on (a, b) f increases on (a, b). f (x) < 0 on (a, b) f decreases on (a, b). f (x) = 0 at x = c the tangent at the point (c, f (c)) is horizontal. Dierentiability and Continuity If f is dierentiable at a, then f is continuous at a. In other words, if f is NOT continuous at a, then f is N OT dierentiable at a. How can a function fail to be dierentiable at a? Three possibilities for a function NOT to have a derivative (i.e., not dierentiable) at a:
3.1 df d The constant multiple rule: dx [cf ] = c dx (cf ) = cf (c is a constant) df dg d The sum or dierence rule: dx [f g] = dx dx (f g) = f g 3.2 df dg d The Product Rule dx [f g] = dx g + f dx (f g) = f g + f g .
d The Quotient Rule dx [ f ] = dx g2 dx ( f ) = f gf g . g g g2 3.3 Derivatives of trigonometric functions (sin x) = cos x, (cos x) = sin x Although the derivatives of other trig functions will be provided, you need to know how to apply these formulas. You also need to learn how to use the Cheat Sheet (copied from the Reference Page 2 of the textbook or posted on the class website, under Section 3.3) to solve problems related to Trigonometry. Besides, you have to understand how to use the unit circle (with cos as the x-axis and sin as the y-axis) to nd the trig values of special angles on dierent quadrants. 3.4 The Chain Rule If g is dierentiable at x and f is dierentiable at g(x), then the composite function F = f g dened by F (x) = f (g(x)) is dierentiable at x and F is given by the product
df
gf dg
1
Remember limx0 sin x x under Section 3.4). 3.5 The method of implicit dierentiation This consists of dierentiating both sides of the implicit function with respect to the independent variable (assumed to be x here) and then solving the resulting equation dy for y = dx . Inverse Trigonometric Functions Although the derivatives of the inverse trig functions will be provided, you need to know how to apply these formulas. 3.6 The Laws of Logarithms: Given a > 0, a = 1, x and y are positive numbers, then 1. log of product = sum of logs: loga (xy) = loga x + loga y 2. log of quotient = dierence of logs: loga ( x ) = loga x loga y y 3. log of power function: loga (xr ) = r loga x You also need to remember loga ax = x, ln ex = x (because ln = loge ), aloga x = x, eln x = x, ln 1 = 0, ln e = 1 and the circle trick: y = loga x means ay = x. The method of logarithmic dierentiation Sometimes, the calculation of the derivative of a function y = F (x) can be simplied by: taking logarithms ln y = ln F (x), using the Laws of Logarithms to simplify the new equation, dierentiating the equation implicitly with respect to x, solving the resulting equation for y . Important Formulas Power Rule: ([g(x)]n ) = n [g(x)]n1 g (x) because (xn ) = n xn1 Trig Rule: [sin(g(x))] = cos(g(x)) g (x) because [sin(x)] = cos(x) [cos(g(x))] = sin(g(x)) g (x) because [cos(x)] = sin(x) Exponential Rule: [ag(x) ] ...