d Derivative of the inverse function and logarithms i E 3 Lecture - PowerPoint PPT Presentation

Section 3.8 d Derivative of the inverse function and logarithms i E 3 Lecture a l l u d b Dr. Abdulla Eid A . College of Science r D MATHS 101: Calculus I Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 1 / 19

Topics d i E 1 Inverse Functions (1 lecture). a l 2 Logarithms. l u d 3 Derivative of the inverse function (1 lecture). b A 4 Logarithmic differentiation (1 lecture). . r D Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 2 / 19

2- Logarithmic Function Consider the exponential function f ( x ) = a x . Question: Does f ( x ) has an inverse? Why? Answer: Yes, by the horizontal line test. d f − 1 ( x ) is called logarithmic function base a and it is denoted by i E a f − 1 ( x ) = log a x l l u d Note: (The fundamental equations) b 1 f ( f − 1 )( x ) = x , so we have a log a x = x . A 2 f − 1 ( f ( x )) = x , so we have log a a x = x . . r D x = a y log a x = y if and only if � �� � � �� � exponential form logarithmic form If a = e = 2.718281828 . . . (Euler number), then we simply write log e as ln “ell en“ and it is called the natural logarithm . Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 3 / 19

Properties of Logarithms 1 log a ( m · n ) = log a m + log a n . d 2 log a ( m n ) = log a m − log a n . i E 3 log a m r = r log a m . a 4 log a 1 = 0. l l u 5 log a a = 1. d b 6 (change of bases) log a m = log b m log b a . A . r D Exercise 1 Use the fundamental equations to prove these six properties of the logarithms. Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 4 / 19

Example 2 (Expansion) Write the following expression as sum or difference of logarithms d i 1 ln ( x wz 2 ) = ln x − ln ( wz 2 ) =ln x − ( ln w + ln z 2 ) =ln x − ln w − 2 ln z . E x + 5 ) 4 = 4 ln ( x + 1 2 ln ( x + 1 a x + 5 ) =4 ( ln ( x + 1 ) − ln ( x + 5 )) . l l √ x ( x 2 )( x + 3 ) 4 ) = ln √ x − ln x 2 − ln ( x + 3 ) 4 = u 3 ln ( d b 1 2 − 2 ln x − 4 ln ( x + 3 ) = 1 2 ln x − 2 ln x − 4 ln ( x + 3 ) = ln x A − 3 2 ln x − 4 ln ( x + 3 ) . . r D Exercise 3 Write each of the following expression as sum or difference of logarithms: � (2) log 2 ( x 5 (3) log ( x 2 z (1) log 3 ( 5 · 7 x + 1 4 ) y 2 ) wy 2 ) (4) ln x − 2 . Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 5 / 19

Example 4 d i E Write each of the following logarithm in terms of natural logarithm. a l l 1 log 3 x = ln x ln 3 . u d 2 log 6 7 = ln 7 ln 6 . b A 3 log 2 y = ln y ln 2 . . r D Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 6 / 19

The derivative of the inverse function Strategy: � � d f − 1 ( x ) Goal: We want to find . dx d i E Write y = f − 1 ( x ) , we want to find y ′ a l l u f ( y ) = f ( f − 1 ( x )) d b f ( y ) = x A f ′ ( y ) · y ′ = 1 . r 1 1 D y ′ = f ′ ( y ) = f ′ ( f − 1 ( x )) Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 7 / 19

Geometric Interpretation * Note that � = 1 d � f − 1 ( x ) f ′ ( f − 1 ( x )) dx so the slope of f − 1 is reciprocal to the slope of f . Geometrically, d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 8 / 19

Example 5 � � Let f ( x ) = x 3 − 3 x 2 − 1. Find f − 1 ( x ) d d dx ( f ( x )) and at the point dx ( 3, − 1 ) Solution: d i E d dx ( f ( x )) = 3 x 2 − 6 x a l l u d dx ( f ( x )) ( 3, − 1 ) = 3 ( 3 ) 2 − 6 ( 3 ) = 9 d b A . r D � = d 1 � f − 1 ( x ) f ′ ( y ) dx 1 = 3 y 2 − 6 y 3 ( 3 ) 2 − 6 ( 3 ) = 1 1 d � � f − 1 ( x ) ( 3, − 1 ) = dx 9 Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 9 / 19

Exercise 6 Let f ( x ) = x + e x . What is the value of f − 1 ( 1 ) . Find ( f − 1 ) ′ ( 1 ) . d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 10 / 19

Derivative of ln Example 7 d Find dx ( ln x ) . d Solution: i E y = ln x a e y = x l l u e y · y = 1 d b y = 1 A e y . r D y ′ = 1 x Exercise 8 Find y ′ if y = log a x . (Hint: Use the change of base formula to change it to ln) Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 11 / 19

Recall d The Chain Rule i E Theorem 9 a l ( f ( g ( x ))) ′ = f ′ ( g ( x )) · g ′ ( x ) l u d b A ( f ( g ( x ))) ′ = derivative of outer ( inner ) · ( derivative of inner ) . r D Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 12 / 19

Example 10 Find y ′ for each of the following: 1 f ( x ) = ln x 2 = ln x 2 → y ′ = x 2 · 2 x = 2 1 x 2 f ( x ) = ln ( 2 x + 3 ) = ln ( 2 x + 3 ) → y ′ = 1 ( 2 x + 3 ) · 2 d i 3 f ( x ) = x ln x → y ′ = ( 1 ) ln x + x · 1 E x = ln x + 1. a 4 f ( x ) = ln ( ln x ) = ln ( ln x ) → y ′ = ( ln x ) · 1 1 x . l l u 5 f ( x ) = ln ( sin x ) = ln ( sin x ) → y ′ = 1 d ( sin x ) · cos x = cot x . b 6 f ( x ) = sin ( ln x ) = sin ( ln x ) → y ′ = cos ( ln x ) 1 A ( x ) . . r D Exercise 11 Find the derivative of the following functions: 1 y = ln ( csc x − cot x ) ln x 2 y = 1 + ln x 3 y = ln ln ln x Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 13 / 19

Derivative using the properties of Logarithms Example 12 Find the derivative of d i 1 f ( x ) = ln x 2017 E a l Solution: First we re–write the function in terms using the properties of l u the ln to get a simplified function: d b A f ( x ) = 2017 ln x . r D Hence f ′ ( x ) = 20171 x Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 14 / 19

Exercise 13 Using the chain rule, find the derivative of the function of the previous example without using the properties of the ln, i.e., find f ′ ( x ) for f ( x ) = ln ( x 2017 ) d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 15 / 19

Derivative using the properties of Logarithms Example 14 Find the derivative of d � x 3 − 1 i 1 f ( x ) = ln 3 E x 3 + 1 a l Solution: First we re–write the function in terms using the properties of l u d the ln to get a simplified function: b A � 1 � x 3 − 1 3 f ( x ) = ln . x 3 + 1 r D = 1 ln ( x 3 − 1 ) − ln ( x 3 + 1 ) � � 3 Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 16 / 19

Continue... We write the inner function in blue and the outer function in red and we apply the chain rule. d i E derivative of outer ( inner ) · ( derivative of inner ) a l l u d b f ( x ) = 1 ln ( x 3 − 1 ) − ln ( x 3 + 1 ) � � A 3 � � . f ′ ( x ) = 1 1 1 r x 3 − 1 · ( 3 x 2 ) − x 3 + 1 · ( 3 x 2 ) D 3 Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 17 / 19

Exercise 15 Using the chain rule, find the derivative of the function of the previous example without using the properties of the ln, i.e., find f ′ ( x ) for   � x 3 − 1  3 d f ( x ) = ln  x 3 + 1 i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 18 / 19

Example 16 Find d 4 y dx 4 for y = 5 ln x d i Solution: E a y ′ = 51 l l u x d = 5 x − 1 b A y ′′ = − 5 x − 2 . y ′′′ = 10 x − 3 r D y ( 4 ) = − 30 x − 4 = − 30 x 4 Dr. Abdulla Eid (University of Bahrain) Logarithmic Differentiation 19 / 19