Derivatives -----------Warren D. Smith-------------- Definition in terms of limit of slope=rise/run of line thru (x, F(x) ) and (x+h, F(x+h)) as run=h approaches 0: LEIBNITZ NEWTON DEFN AS LIMIT d F(x) F(x+h) - F(x) ------ = F'(x) = lim ------------- d x h-->0 h Idea by Newton: if F(t) is distance traveled so far, at time t, then F'(t) = instantaneous speed at time t. F'(t) = "rate of increase" of F(t). The derivative of F(x) at x is the SLOPE of the curve y=F(x) at (x,F(x)). Note, if F(x) is not continuous at x or has a "corner," the limit will not exist and the derivative will not exist. Example F(x) = |x| has a corner at x=0. F'(x) = sign(x) since |x| has slope=1 for x>0 and slope = -1 for x<0 (sign(x) = +1, -1, or 0) ...but f'(0) really does not exist The "d" notation is Leibnitz's, and the F' notation is Newton's. Newton also used "F dot" (place a dot above the F) instead of F' (pronounced "F prime") and dots are still used today in cases where we are talking about TIME derivatives F'(t), t=time, in physics. Note the "d" here has a SPECIAL MEANING, as an infinitesimal difference, and is NOT some variable which can be "canceled" from top and bottom!! In fact dx and dy should be thought of as denoting a SINGLE infinitesimally small, quantity - the d "glues" itself to whatever comes after it. dx means "infinitesimal change in x." TABLE OF DERIVATIVES (B and c denote constants, G(x) and H(x) denote functions) F(x) F'(x) comments ------------------------------------------------------ const 0 horizontal line, slope=0 x+h - x x 1 ------- = 1 h -2 1/(x+h) - 1/x -1 1/x -x ------------- = ------ h (x+h)x n n-1 x n x (n allowed to be negative or non-integer; above three cases were n=0 and n=1 and n=-1; derivation: n=0,1,2,3,4... by binomial theorem; n=-1,-2,-3,... by chain rule; n=1/2 by special analysis of square-rooting; halve n --> n/2 (squre-rooting) and multiply by x: n --> n+1 repeatedly using chain rule to get n to be arbitrarily close to any real) c G(x) c G'(x) scaling rule (since if stretch y-axis rises stretch but runs stay the same) G(x) + H(x) G'(x) + H'(x) adding rule (since rises add, runs stay same) Increasing function Positive Decreasing function Negative At a smooth max or min Zero! (Not increasing and not decreasing) G(x) H(x) G'(x) H(x) + H'(x) G(x) product rule (derived via picture of rectangle with sides G and H) G(H(x)) G'(H(x)) H'(x) "chain rule" for function composition d G(H(x)) d H(x) = --------- ------ (Here's derivation using Leibnitz notation d H(x) d x via "uncancelling two infinitesimals") G(c x) c G'(c x) stretching the x-axis: Because of chain rule on G(x) & cx. Also because runs are divided by c, while rises stay same. -2 1/G(x) -G(x) G'(x) because of chain rule on 1/x & G(x) G(x) G'(x) H(x) - G(x) H'(x) ---- ----------------------- quotient rule H(x) 2 (because: product rule for H(x) G(x) and 1/H(x)) ln(x) 1/x derivation via limit used trick x+h = x(1+h/x) and used fact ln(1+z) = z + smaller terms when z is very small, i.e. slope of the NATURAL log function at 1, is 1. 1 ln(x) log (x) ------- (because log (x) = ----- , and use B x ln(B) B ln(B) scaling rule) x exp(x)=e exp(x) derivation from exp(x+h)=exp(x)exp(h) and use fact exp(h) = 1+h+smaller terms when h is very small, i.e. slope of exp(h) for h=0 is 1, which is same as ln'(1)=1 from relation between graphs of the inverse functions ln(y) and exp(x) x x x B B ln(B) (because B = exp(x ln(B)) and use chain rule) n x DO NOT get confused about x VERSUS B which have totally different derivative formulas!!! WHEN F IS ONE OF THE 6 TRIG FUNCTIONS: F(x) F'(x) comment ----------------------------------------------------------------------------------- sin(x) cos(x) derivation using sin(x+h)=sin(x)cos(h)+sin(h)cos(x) and sin(h)/h -->1 for h very small (geometry: arc length --> line length) and 1-cos(h) << h for h very small (geometry: use fact cos(h) has maximum at h=0). cos(x) -sin(x) since cos(x), sin(x), -sin(x), -cos(x) all have the same graph, just shifted horizontally by various amounts... The above two (for sin and cos) are the master formulae; the 4 others (below) sin(x) 2 may be derived from them using quotient rule tan(x)=------ sec(x) and simplified using the pythagorean trig identity cos(x) 2 2 sin(x) + cos(x) = 1. cos(x) 2 MNEMONIC: Notice that the "co" functions cot(x)=------ -csc(x) (cosine, cosecant, cotangent) have sin(x) NEGATIVE signs in their derivative formulas. Indeed the formulas for the derivs 1 of the "co" functions are the SAME sec(x)=------ sec(x) tan(x) as the formulas for the deriv of cos(x) the corresponding (non-co) function except that (1) sign has changed and (2) all functions changed to their cofunctions. 1 csc(x)=------ -csc(x) cot(x) MNEMONIC: Definitions of cofunctions are the same sin(x) as the usual definition of non-co trig functions, but with cos's swapped with sin's. ======================================================= Good exercises: Find formulas for the derivatives of x 1/2 x sin(cos(x)) sin(x ) sin(ln(cos(x))) x sin(x) 1/x sin(x)ln(x) sin(x)/ln(x+7) ln(7x) x B Hint: A = exp(B ln(A)) is a good start, you then will be able to use known formulas for derivatives of exp and ln. If this is taken to be the DEFINITION of A to the power B, then B>0 is the B-domain since ln(B) is only defined if B>0. 1/x Plot y = x (what is its DOMAIN?). This curve has a maximum at coordinates x=? and y=?. Can you determine the location of the maximum with the help of derivatives and equation manipulation? Plot y = x ln(x) (what is its DOMAIN?). This curve has a minimum at coordinates x=? and y=?. Can you determine the location of the minimum with the help of derivatives and equation manipulation? ========================================================== The LINEAR APPROXIMATION to F(x) based at x=A is: y = (x-A) F'(A) + F(A) i.e. the line with slope F'(A) and such that when x=A, y=F(A).