Chapter Derivatives Section. An Intuitive Introuction to Derivatives Consier a function: Slope function: Derivative, f ' For each, the slope of f is the height of f ' Where f has a horizontal tangent line, the erivative f ' has a root. Where the graph of f is increasing, the erivative f ' is above the -ais. Where the graph of f is ecreasing, the erivative f ' is below the -ais. Where f has steep slope, the erivative f ' has large magnitue Where f has shallow slope, the erivative f ' has small magnitue Eample (7, page 66) Sketch a graph of the associate slope function f '.
Eample (3, page 66) The graph below is f ', sketch a possible graph of f. Average Velocity: For the function, y f, let s look at the slope of the secant line from a to b. Now the slope of the secant line tells us the average rate change of the function from a to b a h. Suppose s f t y f b f a f a h f a average rate of change b a h where s is the isplacement (irecte istance) from origin at time t an f is the position function (escribes the motion) of the object. In the time interval from t ato t a h, the average rate of change woul tell us the average velocity over this time interval is isplacement average velocity= time s f t f t f a h f a t t t h
Consier the position function s t 4.9t 30t 0, what is the average rate of change for each of the following: Time Interval,3,.5,.,.0,.00 Average Velocity What happens to the slope of the secant line as h 0? The instantaneous rate of change of a function is the slope of f at c. This is also the erivative of f at c. Let s look at a graph of the function an the various secant lines from above. What woul the slope of the tangent line be? What woul you estimate the instantaneous velocity at t? 3
Eample (#5, page 67) Every morning Lyna takes a thirty-minute jog in Central Park. Suppose her istance s in feet from the oak tree on the north sie of the park t minutes after she begins her jog is given by the function st, shown below on the left an suppose she jogs on a straight path leaing into the park from the oak tree. a. What was the average rate of change of Lyna s istance from the oak tree over the entire thirty-minute jog? What oes that mean in the real worl? b. On which ten-minute interval was the average rate of change of Lyna s istance from the oak tree the greatest: the first ten minutes, the secon ten minutes the last ten minutes? c. Use the graph of st to estimate Lyna s average velocity uring the 5-minute interval from t 5 to t 0. What oes the sign of this average velocity tell us?. Approimate the times at which Lyna s (instantaneous) velocity was equal to zero. What is the physical significance of these times? e. Approimate the time intervals uring Lyna s jog that her (instantaneous) velocity was negative. What oes a negative velocity mean in terms of this physical eam? 4
Section. Formal Definition of the Derivative Definition. The Derivative of a Function at a Point The erivative c of a function of f is the number f c h f f ' c lim h 0 h provie the limit eists. or equivalently f ' c f z f c lim z c zc Eample: Fin the erivative of the function f at 4. Definition. The Derivative of a Function The erivative of a function of f is the function f h f f ' lim h 0 h The omain of ' Eample: Fin the erivative of f. f ' efine by f z f or equivalently f ' c lim z z f is the set of values for which the efining limit of f ' eists.. Definition.3 Differentiability at a Point A function f is ifferentiable at c if f c h f c lim h 0 h eists. 5
Definition.4 One-Sie Differentiability at a Point The left erivative an right erivative of a function f at a point c are respectively, equal to the following, if they eist: ' f c h f c ' f c h f c f c lim, f lim c h0 h h0 h Theorem.5 Differentiability Implies Continuity If f is ifferentiable at c, then f is continuous at c. Theorem (Alternative Version) Not Continuous Implies Not Differentiable If f is not continuous at a, then f is not ifferentiable at a. When is a Function Not Differentiable at a Point? A function f is not ifferentiable at cif at least one of the following conitions hols: a. f is not continuous at c. b. f has a corner at c. c. f has a vertical tangent at c. Eample: Show that the function y has no erivative at 0. 6
Note: If f ' oes not eist At a corner lim f ' l an lim f ' l c c c At a vertical tangent: o (Cusp) lim f ' an lim f ' o lim f ' c an lim f ' c c where l l. (or lim f ' c (or both equal ) an lim c f ' ) Theorem.6 Equation of the Tangent Line to a Function at a Point The tangent line to the graph of a function f at a point cis efine to be the line passing through, ' f ' c eists. This line has equation c f c with slope f c, provie that the erivative ' y f c f c c Definition.7 Local Linearity If f has a well-efine erivative function f ' c at a point c, then for values of near c, the f can be approimate by the tangent line to f at cwith the linearization of f aroun c by ' f f c f c c y f Other notation: f ' y' f 7
Section.3 Rules for Calculating Basic Derivatives Theorem.8 Derivatives of Constant, Ientity, an Linear Functions For any real numbers k, m, an b. k 0. 3. m b m Eample: Fin the erivative of the following:. 3 f e. 3. f 7 99 Theorem.9 Power Rule k k k For any nonzero rational number k, Proof: Let k be a positive integer. Then k k k f z f z lim lim z z z z z z z z lim z z k k k k lim z z z z k k k k k k Eample: Fin the erivative for each of the following: f. 8
5. f 3. f 4. f Theorem.0 Constant Multiple Rule If f is a ifferentiable function of, an k is a constant, then f kf k kf ' Eample: Fin the erivative for each of the following: f 7. 4. f 3. f 5 Theorem.0 Sum Rule If f an g are ifferentiable functions of, then f g f g Eample: Fin the erivative of f 5 6. 9
f 3. 3. y 3 8 s s s 4s Theorem. Prouct Rule If f an g are ifferentiable at, then f g f ' g f g ' Other notation: v u uv u v Eample: Fin the erivative for each of the following: f 3.. f 0 3 Theorem. The Quotient Rule If f an g are ifferentiable at, then the erivative of f / g at eists provie g 0 an f g f ' f g ' g g 0
u v v u u Other notation: v v One more way: high low low low high high low Eample Fin the erivative for each of the following: 34. y f 6. Fin y for each of the following: /3 y.. 6 y 3 y 3 3. 3 4. 3 y 5 4 5. y
Higher-Orer Derivatives Definition: Higher-Orer Derivatives Assuming f can be ifferentiate as often as necessary, the secon erivative of f is\ y y f " f y" D f For integers n, the nth erivative is n n y n f f n f 3 6 4 Eample: Fin the following:. f '. f " 3. f '" Eample: Fin the following if f 3. 3. 3 3 3. 3 3
Section.4 The Chain Rule an Implicit Differentiation Theorem. The Chain Rule Suppose f u is a composition of functions. Then for all values of at which u is ifferentiable at an f is ifferentiable at u, the erivative of f with respect to is equal to the prouct of the erivative of f with respect to u an the erivative of u with respect to. In prime notation, we write it as In Leibniz notation we write it at ' ' f u f u u f f u u Note we can translate the following in wors; ifferentiate the outsie function an evaluate it at the insie function, then multiple by the erivative of the insie function. y f u f ' u u ' Eample: Fin the erivatives of the following functions: y. 6. y 5 7 4 3 y 3. /3 5 3 3 3 00 y 4 4. 3
5. 9 y 4 3 6. f 7. y 34 3 7 8. y 3 3 5 Eample: Determine an equation of the line tangent to the graph of,. y 5 at the point Implicit Differentiation In the previous sections, we have mostly ealt with equations of the form y f that epresses y eplicitly in terms of the variable. (In other wors, y is the epenent variable an is a function of only the inepenent variable.) However, what happens if we have an implicit relation between the variables y an. Look at the following eamples: 3 3 y 8 y y y 3 3y 5y 4
Implicit Differentiation. Differentiate both sies of the equation with respect to, treating y as a ifferentiable function of.. Collect the terms with y on one sie of the equation. 3. Solve for y. Eample: Fin y for y r. 3 3 Eample: Fin y for y 8y. Eample: Fin y for y y 3 Eample: Fin y for 3y 5y Eample: Use implicit ifferentiation to fin y an then fin y for y y Eample: For the equation 9 y, fin the line that is tangent the curve at,3 5
Section.5 Derivatives of Eponential an Logarithmic Functions Theorem.3 Derivatives of Eponential Functions For any constant k, any constant b > 0 with b, an all,. e e. b ln b b 3. k k e ke Eample: Fin the erivatives of the following:. y 5e. y e 6 3. y e Inverse Properties for ln. e for 0 e an ln, an lne. y ln if an only if e 3. For real numbers an b 0, 4. y y for all. lnb b e e lnb 5. y 7 Theorem.4 Rates of Change an Eponential Functions f kf for some constant k if an only if f is an eponential function of the form ' k Ae f 6
Eample: Fin the erivative of y ln. Eample: Fin the erivative of. y ln3 3/. y ln. 3. y ln 4. y log 7 Theorem.5 Derivative of Logarithmic Functions For any constant b 0 with b an all appropriate values of,. ln, 0. ln, 0 3. logb lnb Suggestions: e u u u e an lnu u u 7
Logarithmic Rules b0 b an positive real numbers an y the following relations hol: For any base. log y log log y b b b. log log log y y log ylog 3. b 4. logb b b b b b y inclues log b log b Logarithmic Differentiation. Take the natural logarithm of both sies of the equation.. Simplify using the laws of logarithms. 3. Take the erivatives of both sies with respect to. 4. Solve for y. Eample: Fin y for the following:. y 3 7 6. y Theorem.6 Derivatives of Inverse Function If f an f are inverse functions an both are ifferentiable, then for all appropriate values of, f ' f ' f 8
f Eample: Given 36,4. f at the point for, fin the slope of the line tangent to the graph of Section.6 Derivatives of Trigonometric an Hyperbolic Functions Theorem.7 Derivatives of Trigonometric Functions sin cos cos sin tan sec cot csc sec tan sec csc csc cot Eample: sin Show cos Eample: tan sec Show Eample: Fin the erivative for each of the following. y 0 3cos. f sin cos sec 9
3. y cos sin cos 4. y sin 5. y sin Eample: 4 4 4 Fin y y y if y 9cos. Eample: Fin the horizontal tangents for y cos on 0. Eample: Fin the erivative of y arcsin. Eample: Fin the erivative of y arctan. 0
Eample: Fin the erivative of y arctan ln. Eample: Fin the erivative of y arcsin 6 Theorem.8 Derivative of Inverse Trigonometric Functions. arcsin,. arctan 3. arcsec, Combining two eponential functions forms hyperbolic functions: Hyperbolic Functions: (See page 8 for graphs) e e Hyperbolic sine: sinh Hyperbolic cosecant: csch sinh e e e e Hyperbolic cosine: cosh Hyperbolic secant: sech cosh e e Hyperbolic tangent: Hyperbolic sinh e e cosh e e tanh cotangent: coth cosh e e sinh e e Trigonometric functions are sometimes referre to as circular functions. If we look at the unit circle, y ( y, ) cos,sin, where is measure in raians. Also, then any point recall that cos sin. Hyperbolic functions get their name because if we look at the right han sie of the hyperbola, y, then any point (, y) (cosh t,sinh t). Note the first ientity for hyperbolic functions is cosh tsinh t. Note in the iagram below, t oes not represent the angle. However, it turns out t represents twice the area of the sector boune by the hyperbola, -ais,
an line. Hyperbolic functions are use for instance to moel the wire hanging between two poles (assuming it is attache to both polls at the same height), then the shape of the curve can be y c acosh / a calle a catenary. moele with cos,sin cosh t,sinh t y y Ientities for Hyperbolic Functions cosh sinh sinh sinh cosh cosh cosh Eample: Fin the erivative y for. y sinh tanh sech cosh cosh sinh cosh sinh. y tanh sinh 3. y
Theorem.0 Derivatives of Hyperbolic Functions For all real numbers, u. sinh u cosh u u. cosh u sinh u u 3. tanh u sech u Eample: Fin the erivative of the following: 3. y sinh. y 3tanh e 3. y tanh cosh Inverse Hyperbolic Functions y sinh sinh y sinh ln cosh cosh, an 0 y y y y tanh tanh y Eample: Fin y for y sinh. cosh ln tanh ln 3
Theorem. Derivatives of Inverse Hyperbolic Functions For all for which the following are efine,. sinh. cosh, 3. tanh, Eamples: Fin the erivative of each of the following: 3 y sinh.. sinh y cosh 4