Table of Contents Derivatives of Logarithms

Transcription

1 Derivatives of Logarithms- Table of Contents Derivatives of Logarithms Arithmetic Properties of Logarithms Derivatives of Logarithms Example Example 2 Example 3 Example 4 Logarithmic Differentiation Example 5 Example 6 Example 7 Negative x A Short Quiz

2 Derivatives of Logarithms-2 Arithmetic Properties of Logarithms Recall that the logarithm functions satisfy very important arithmetic laws: If a an b are positive numbers, an c is a positive number not equal to, we have: log c a = ln a ln c ln(ab) = ln a + ln b log c (ab) = log c a + log c b ln ( ) ( ) a a = ln a ln b log b c = log b c a log c b ln (a b) ( = b ln a log c a b) = b log c a

3 Derivatives of Logarithms-3 Derivatives of Logarithms The natural logarithm function was efine to the inverse of the exponential function e x, an the base a logarithms were efine to be the inverse of the exponential function a x, so we have the Cancellation Laws e ln x = x an a ln x = x, so if we let y = ln x or y = log a x we have e y = x or a y = x. Using the metho of implicit ifferentiation, we get: (e y ) = e y y = (x) = or (a y ) = (ln a)a y y = (x) =, so y = e y = x or y = (ln a)a y = (ln a)x Thus x (ln x) = x an x ( loga x ) = (ln a)x

4 Derivatives of Logarithms-4 We may get more general formulas by using the Chain Rule: x (ln f(x)) = f (x) f(x) an x ( loga f(x) ) = f (x) (ln a)f (x) = f (x) ln a f(x) The quantity f (x) f(x) applications. is calle the relative rate of change of the function f, an is very important in practical

5 Derivatives of Logarithms-5 Example : Fin the erivative of y = ln(x 5 + x 3 + ). y = x 5 + x 3 + (x5 + x 3 + ) = x 5 + x 3 + (5x4 + 3x 2 ) = x2 (5x 2 + 3) x 5 + x 3 + Example 2: Fin x (sin(ln(x)) = cos(ln x) x x (sin(ln(x)). = cos(ln x) x

6 Derivatives of Logarithms-6 Example 3: Fin (ln (sin(ln(x))). x (ln (sin(ln(x))) = x sin (ln(x)) (sin (ln(x))) = sin(ln(x)) cos(ln(x))(ln x) = sin(ln(x)) cos(ln(x)) x = cos(ln(x)) x sin(ln(x) Note that the omain of the erivative function is much larger than that of the original function!

7 Derivatives of Logarithms-7 Example 4: Fin x log 6(sin x + cos x). ln(sin x + cos x) log 6 (sin x + cos x) =,so ln 6 x log 6(sin x + cos x) = ln(sin x + cos x) = ln 6 x ln 6 sin x + cos x (sin x + cos x) = ln 6 sin x + cos x (cos x sin x) = cos x sin x (ln 6)(sin x + cos x)

8 Derivatives of Logarithms-8 Logarithmic Differentiation One of the most important uses of the natural logarithm function is in the computation of erivatives of functions which are mae up of proucts, quotients an powers of more elementary functions. We use the three basic arithmetic properties of the logarithm to simplify the function. Example 5: Fin y if y = (x + ) 4 (x ) 3 (x + 4) 7 Taking logarithms of both sies of the equation, we get ( ) (x + ) 4 ln y = ln (x ) 3 (x + 4) 7 or ln y = 4ln(x + ) 3ln(x ) 7ln(x + 4) which we now ifferentiate: y y = 4 x + 3 x 7 x + 4 which we nee only simplify slightly to get y in a usable form: [ 4 y = y x + 3 x 7 ] x + 4

9 Derivatives of Logarithms-9 Example 6: Fin y if y = x4 π x sin 5 (3x) (x ) 3 (x + 2) 7.5 Taking logarithms of both sies of the equation, we get ( ) x 4 π x sin 5 (3x) ln y = ln (x ) 3 (x + 2) 7.5 or ln y = 4lnx + x ln π + 5lnsin(3x) 3ln(x ) 7.5ln(x + 2) which we now ifferentiate: y y = 4 3 cos(3x) + ln π + 5 x sin(3x) 3 x 7.5 x + 2 which we nee only simplify slightly to get y in a usable form: [ 4 y 3 = y + ln π + 5 cot(3x) x x 7.5 ] x + 2

10 Derivatives of Logarithms-0 Example 7: Sketch the graph of y = x x Since, by efinition, x x = ( e ln x) x = e x ln x, the function is only efine for x>0, an must always be positive. We have ln y = ln (x x ) = x ln x, so y y = (x ln x) = (x) (ln x) + x(ln x) = () ln x + x x = + ln x, soy = x x ( + ln x). The sign of y epens only on that of + ln x, which is 0 if ln x = orx = e = e If 0 <x< e we have y < 0 an if e <xwe have y > 0. 4 y x

11 Derivatives of Logarithms- Negative x There will be occasions when we wish to apply logarithms an eal with negative values of the variables concerne. Of course, ln x is unefine if x 0. However, ln x is efine if x<0. Let us then fin the erivative of ln x for non-zero x. If x>0, it is of course x. If x<0, then x = x, soln x =ln( x), an we can apply the Chain Rule: x (ln( x)) = x x ( x) = x ( ) = x, so we have the important formula x (ln x ) = x if x 0