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# Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 handout)

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The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.

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### Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 handout)

1. 1. Section 3.3 Derivatives of Exponential and Logarithmic Functions V63.0121.041, Calculus I New York University October 25, 2010 Announcements Midterm is graded. Please see FAQ. Quiz 3 next week on 2.6, 2.8, 3.1, 3.2 Announcements Midterm is graded. Please see FAQ. Quiz 3 next week on 2.6, 2.8, 3.1, 3.2 V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 2 / 34 Objectives Know the derivatives of the exponential functions (with any base) Know the derivatives of the logarithmic functions (with any base) Use the technique of logarithmic diﬀerentiation to ﬁnd derivatives of functions involving roducts, quotients, and/or exponentials. V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 3 / 34 Notes Notes Notes 1 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
2. 2. Outline Recall Section 3.1–3.2 Derivative of the natural exponential function Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Diﬀerentiation The power rule for irrational powers V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 4 / 34 Conventions on power expressions Let a be a positive real number. If n is a positive whole number, then an = a · a · · · · · a n factors a0 = 1. For any real number r, a−r = 1 ar . For any positive whole number n, a1/n = n √ a. There is only one continuous function which satisﬁes all of the above. We call it the exponential function with base a. V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 5 / 34 Properties of exponential Functions Theorem If a > 0 and a = 1, then f (x) = ax is a continuous function with domain (−∞, ∞) and range (0, ∞). In particular, ax > 0 for all x. For any real numbers x and y, and positive numbers a and b we have ax+y = ax ay ax−y = ax ay (negative exponents mean reciprocals) (ax )y = axy (fractional exponents mean roots) (ab)x = ax bx V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 6 / 34 Notes Notes Notes 2 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
3. 3. Graphs of various exponential functions x y y = 1x y = 2xy = 3x y = 10x y = 1.5x y = (1/2)xy = (1/3)x y = (1/10)x y = (2/3)x V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 7 / 34 The magic number Deﬁnition e = lim n→∞ 1 + 1 n n = lim h→0+ (1 + h)1/h V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 8 / 34 Existence of e See Appendix B We can experimentally verify that this number exists and is e ≈ 2.718281828459045 . . . e is irrational e is transcendental n 1 + 1 n n 1 2 2 2.25 3 2.37037 10 2.59374 100 2.70481 1000 2.71692 106 2.71828 V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 9 / 34 Notes Notes Notes 3 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
4. 4. Logarithms Deﬁnition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga(x1 · x2) = loga x1 + loga x2 (ii) loga x1 x2 = loga x1 − loga x2 (iii) loga(xr ) = r loga x V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 10 / 34 Graphs of logarithmic functions x y y = 2x y = log2 x (0, 1) (1, 0) y = 3x y = log3 x y = 10x y = log10 x y = ex y = ln x V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 11 / 34 Change of base formula for logarithms Fact If a > 0 and a = 1, and the same for b, then loga x = logb x logb a Proof. If y = loga x, then x = ay So logb x = logb(ay ) = y logb a Therefore y = loga x = logb x logb a V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 12 / 34 Notes Notes Notes 4 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
5. 5. Upshot of changing base The point of the change of base formula loga x = logb x logb a = 1 logb a · logb x = (constant) · logb x is that all the logarithmic functions are multiples of each other. So just pick one and call it your favorite. Engineers like the common logarithm log = log10 Computer scientists like the binary logarithm lg = log2 Mathematicians like natural logarithm ln = loge Naturally, we will follow the mathematicians. Just don’t pronounce it “lawn.” V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 13 / 34 Outline Recall Section 3.1–3.2 Derivative of the natural exponential function Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Diﬀerentiation The power rule for irrational powers V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 14 / 34 Derivatives of Exponential Functions Fact If f (x) = ax , then f (x) = f (0)ax . Proof. Follow your nose: f (x) = lim h→0 f (x + h) − f (x) h = lim h→0 ax+h − ax h = lim h→0 ax ah − ax h = ax · lim h→0 ah − 1 h = ax · f (0). To reiterate: the derivative of an exponential function is a constant times that function. Much diﬀerent from polynomials! V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 15 / 34 Notes Notes Notes 5 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
6. 6. The funny limit in the case of e Remember the deﬁnition of e: e = lim n→∞ 1 + 1 n n = lim h→0 (1 + h)1/h Question What is lim h→0 eh − 1 h ? Answer If h is small enough, e ≈ (1 + h)1/h . So eh − 1 h ≈ (1 + h)1/h h − 1 h = (1 + h) − 1 h = h h = 1 So in the limit we get equality: lim h→0 eh − 1 h = 1 V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 16 / 34 Derivative of the natural exponential function From d dx ax = lim h→0 ah − 1 h ax and lim h→0 eh − 1 h = 1 we get: Theorem d dx ex = ex V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 17 / 34 Exponential Growth Commonly misused term to say something grows exponentially It means the rate of change (derivative) is proportional to the current value Examples: Natural population growth, compounded interest, social networks V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 18 / 34 Notes Notes Notes 6 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
7. 7. Examples Examples Find derivatives of these functions: e3x ex2 x2 ex Solution d dx e3x = 3e3x V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 19 / 34 Outline Recall Section 3.1–3.2 Derivative of the natural exponential function Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Diﬀerentiation The power rule for irrational powers V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 20 / 34 Derivative of the natural logarithm function Let y = ln x. Then x = ey so ey dy dx = 1 =⇒ dy dx = 1 ey = 1 x We have discovered: Fact d dx ln x = 1 x x y ln x 1 x V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 21 / 34 Notes Notes Notes 7 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
8. 8. The Tower of Powers y y x3 3x2 x2 2x1 x1 1x0 x0 0 ln x x−1 x−1 −1x−2 x−2 −2x−3 The derivative of a power function is a power function of one lower power Each power function is the derivative of another power function, except x−1 ln x ﬁlls in this gap precisely. V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 22 / 34 Outline Recall Section 3.1–3.2 Derivative of the natural exponential function Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Diﬀerentiation The power rule for irrational powers V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 23 / 34 Other logarithms Example Use implicit diﬀerentiation to ﬁnd d dx ax . Solution Let y = ax , so ln y = ln ax = x ln a Diﬀerentiate implicitly: 1 y dy dx = ln a =⇒ dy dx = (ln a)y = (ln a)ax Before we showed y = y (0)y, so now we know that ln a = lim h→0 ah − 1 h V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 24 / 34 Notes Notes Notes 8 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
9. 9. Other logarithms Example Find d dx loga x. Solution Let y = loga x, so ay = x. Now diﬀerentiate implicitly: (ln a)ay dy dx = 1 =⇒ dy dx = 1 ay ln a = 1 x ln a Another way to see this is to take the natural logarithm: ay = x =⇒ y ln a = ln x =⇒ y = ln x ln a So dy dx = 1 ln a 1 x . V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 25 / 34 More examples Example Find d dx log2(x2 + 1) Answer V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 26 / 34 Outline Recall Section 3.1–3.2 Derivative of the natural exponential function Exponential Growth Derivative of the natural logarithm function Derivatives of other exponentials and logarithms Other exponentials Other logarithms Logarithmic Diﬀerentiation The power rule for irrational powers V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 27 / 34 Notes Notes Notes 9 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
10. 10. A nasty derivative Example Let y = (x2 + 1) √ x + 3 x − 1 . Find y . Solution We use the quotient rule, and the product rule in the numerator: y = (x − 1) 2x √ x + 3 + (x2 + 1)1 2(x + 3)−1/2 − (x2 + 1) √ x + 3(1) (x − 1)2 = 2x √ x + 3 (x − 1) + (x2 + 1) 2 √ x + 3(x − 1) − (x2 + 1) √ x + 3 (x − 1)2 V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 28 / 34 Another way y = (x2 + 1) √ x + 3 x − 1 ln y = ln(x2 + 1) + 1 2 ln(x + 3) − ln(x − 1) 1 y dy dx = 2x x2 + 1 + 1 2(x + 3) − 1 x − 1 So dy dx = 2x x2 + 1 + 1 2(x + 3) − 1 x − 1 y = 2x x2 + 1 + 1 2(x + 3) − 1 x − 1 (x2 + 1) √ x + 3 x − 1 V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 29 / 34 Compare and contrast Using the product, quotient, and power rules: y = 2x √ x + 3 (x − 1) + (x2 + 1) 2 √ x + 3(x − 1) − (x2 + 1) √ x + 3 (x − 1)2 Using logarithmic diﬀerentiation: y = 2x x2 + 1 + 1 2(x + 3) − 1 x − 1 (x2 + 1) √ x + 3 x − 1 Are these the same? Which do you like better? What kinds of expressions are well-suited for logarithmic diﬀerentiation? V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 30 / 34 Notes Notes Notes 10 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010
11. 11. Derivatives of powers Question Let y = xx . Which of these is true? (A) Since y is a power function, y = x · xx−1 = xx . (B) Since y is an exponential function, y = (ln x) · xx (C) Neither x y 1 1 Answer (A) This can’t be y because xx > 0 for all x > 0, and this function decreases at some places (B) This can’t be y because (ln x)xx = 0 when x = 1, and this function does not have a horizontal tangent at x = 1. V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 31 / 34 Derivatives of power functions with any exponent Fact (The power rule) Let y = xr . Then y = rxr−1 . Proof. y = xr =⇒ ln y = r ln x Now diﬀerentiate: 1 y dy dx = r x =⇒ dy dx = r y x = rxr−1 V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 33 / 34 Summary Derivatives of logarithmic and exponential functions: y y ex ex ax (ln a) · ax ln x 1 x loga x 1 ln a · 1 x Logarithmic Diﬀerentiation can allow us to avoid the product and quotient rules. V63.0121.041, Calculus I (NYU) Section 3.3 Derivs of Exp and Log October 25, 2010 34 / 34 Notes Notes Notes 11 Section 3.3 : Derivs of Exp and LogV63.0121.041, Calculus I October 25, 2010