Announcements Topics: - sections 4.5 and 5.1-5.5 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the SCHEDULE + HOMEWORK link)
Basic Differentiation Rules All rules are proved using the definition of the derivative: df dx = f '(x) = lim h 0 f (x + h) f (x) h The derivative exists (i.e. a function is differentiable) at all values of x for which this limit exists.
The Constant Function Rule If where is a constant, then f (x) = k, k f '(x) = 0. f (x) = 3 f '(x) = 0 y m=0 x
The Power Rule If where then f (x) = x n, n R, f '(x) = nx n 1. Differentiate the following. (a) g(x) = x 100 (b) f (x) = x (c) (d) h(x) = 1 x 6 s(t) = t
Let k Then The Constant Multiple Rule be a constant. d dx k f (x) [ ] = k d dx Find the derivative of each. (a) (b) [ f (x) ]. f (x) = 3x 7 g(x) = 5 x
The Sum/Difference Rule provided f and g are differentiable functions. Examples: Differentiate. (a) [ f (x) ± g(x)]'= f '(x) ± g'(x) f (x) = 3x 4 + 5x 2 10 (b) g(x) = 5x + x 5 π 2
Using the Derivative to Sketch the Graph of a Function Sketch the graph of F(x) = x + 1 x. y x
Derivative of the Natural Exponential Function Definition: The number e is the number for which y f (x) = e x lim h 0 e h 1 h =1 x Natural Exponential Function: f (x) = e x
Derivative of the Natural Exponential Function Note: This definition states that the slope of the tangent to the curve at (0,1) is exactly 1, i.e. f '(0) = lim h 0 e 0+h e 0 h y m =1 f (x) = e x x = lim h 0 e h 1 h =1
Derivative of the Natural Exponential Function If f (x) = e x, then ʹ f (x) = e x. In words: The slope of the tangent line to the curve f (x) = e x at the point P is equal to the value of the function at P. y f (x) = e x P (x, e x )... slope = e x (1, e)... slope = e (0, 1)... slope =1 x
The Product Rule [ f (x) g(x)]'= f '(x) g(x) + f (x) g'(x) provided f and g are differentiable functions. Find the critical numbers of f (x) = x 4 e x.
The Quotient Rule " $ # f (x) g(x) % ''= & f '(x) g(x) f (x) g'(x) [g(x)] 2 provided and are differentiable and f g g(x) 0. Determine where the graph of the function q(x) = x x 2 + 8 has horizontal tangents.
Chain Rule [ f (g(x))]'= f '(g(x)) g'(x) derivative of the outer function evaluated at the inner function times the derivative of the inner function Differentiate the following. (a) (b) f (x) = (4x 3 +1) 10 g(x) = 1 x 2 + e 3x
Derivatives of General Exponential Functions If then f (x) = a x, f '(x) = a x ln a. Differentiate. (a) (b) (c) f (x) = e x g(x) = 7 x + x 7 h(x) = 2 5x 2 +1
If Derivatives of Logarithmic Functions f (x) = log a x, then f '(x) = 1 x ln a. Example 1: Differentiate. (a) f (x) = ln x (b) g(x) = log 4 (x 2 + 5x + 6) Example 2: Determine the equation of the tangent line to the curve f (x) = ln x at the point P(1,0). x
Derivatives of Trigonometric Functions 2 1 y = sin x y = sin x 5-4 -3-2 -1 0 1 2 3 4 5-1 -2 2 1 y = cos x 5-4 -3-2 -1 0 1 2 3 4 5-1 -2
Derivatives of Trigonometric Functions (sin x)'= cos x (cos x)'= sin x (tan x)'= sec 2 x Find the derivative of each. (a) f (x) = cos(e 3x 2 ) (b) g(x) = csc x (c) h(x) = tan x 4 + tan 4 x
Derivatives of Inverse Trig Functions d dx (arcsin x) = 1 1 x 2 d dx (arctan x) = 1 1+ x 2 Differentiate. (a) (b) f (x) = arctan x 5 ( ) + arctan ( 5 x) g(x) = ln(arcsin(5x 3 +1))
Implicit Differentiation Using implicit differentiation, determine for y 3 + x 2 = e xy. y "
Logarithmic Differentation Using logarithmic differentiation, find the derivative of h(x) = (x +1)ex x 3 sin x
Related Rates The concentration of a pollutant (measured in grams per cubic metre) at a location x metres away from the source is given by c(x) = 0.28e 0.25x2 An observer is located 5m from the source. How does the concentration change as she runs away from the source at a speed of 4.8m/s?