Calculus Question

Calculus: Learning Worksheets Chapter 3 Name ________________________________ Date ______________ Class ____________ Section 3-2 Derivatives of Exponential and Logarithmic Functions Goal: To find the first derivative of functions that are exponential or logarithmic in nature. Definition: Derivatives of Exponential and Logarithmic Functions For b  0, b  1, d x e  ex dx d x b  b x ln b dx d 1 ln x  dx x d 1 1 logb x    dx ln b  x  In Problems 1–5, Find the first derivative for the given function. 1. f ( x) = 7e x + 4 x5 − 3 x + 21 2. f ( x) = −5e x − 7 ln x − 13 x + 4 95 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 3. f ( x)  4 x3  2 x 2  5 x  ln x5  e x 4. y  10 x  e3  5 x 5. y  log 7 x  3 x5  4 x Chapter 3 96 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 6. Chapter 3 Given the function f ( x) = 12e x + 5, find the equation of the line tangent to the graph of the function at x  0. 97 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 7. Chapter 3 Given the function f ( x)  ln x5  2, find the equation of the line tangent to the graph of the function at x  e. 98 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 3 Name ________________________________ Date ______________ Class ____________ Section 3-3 Derivatives of Products and Quotients Goal: To find the first derivative of functions that are written as products or quotients. Theorem 1: Product Rule If y  f ( x)  F ( x) S ( x) and if F  ( x) and S  ( x) exist, then f  ( x)  F ( x) S  ( x)  S ( x) F  ( x). Theorem 2: Quotient Rule If y  f ( x)  T ( x) and if T  ( x) and B  ( x) exist and B ( x)  0 , then B( x) f  ( x)  B( x)T  ( x)  T ( x) B  ( x)  B( x)2 . In Problems 1–10, find the first derivative and simplify. 1. f ( x) = 7 x 6 (3 x3 + 14 x − 10) 99 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 2. f ( x)  (4 x 2  7)(2 x 2  9) 3. f ( x) = Chapter 3 17 x 5 x 2 + 13 100 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 3 2x  5 4. f ( x)  5. f ( x) = 7e x ( 7 ln x ) 3x 2  5 ( ) 101 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 6. y  (2 x3  5 x 2 )( x 2  2 x) 7. y Chapter 3 x 2  3x  4 4 x2  3 102 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 8. f ( x) = 7 x log5 x 9. f ( x)  Chapter 3 84 x x2  4 103 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 10. f ( x)  Chapter 3 (2 x3  5)(3x 2  1) x2  7 104 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 3 Name ________________________________ Date ______________ Class ____________ Section 3-4 The Chain Rule Goal: To find the first derivative of functions that are written as composite functions. Definition: Composite Function A function m is a composite of functions f and g if m( x )  f  g ( x )  . The domain of m is the set of all numbers x such that x is in the domain g and g ( x) is in the domain of f. Theorem 1: General Power Rule If u ( x) is a differentiable function, n is any real number, and y  f ( x)  u ( x)  , then n n1 f  ( x )  n u ( x )  Theorem 2: u  ( x). Chain Rule If y  f (u ) and u  g ( x) define the composite function y  m( x)  f  g ( x)  , then dy dy du  dx du dx 1. provided dy du and exist. du dx Find m( x)  f  g ( x)  if f (u ) = u 4 and g ( x) = 2 x 2 − 8 x + 15. 105 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 2. Chapter 3 Find m( x)  f  g ( x)  if f (u )  ln u and g ( x)  4 x 2  7 x  35. 3. Given y = ( x5 − 7 x 2 + 2 x − 21)5 , write this composite function in the form y  f (u ) and u  g ( x). 3 4. Given y  e5 x  4 x  23 , write this composite function in the form y  f (u ) and u  g ( x). In Problems 5–10, find the first derivative for the given functions. 5. f ( x) = (3 x 4 − 5 x + 13)5 106 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 6. f ( x) = 7 ln(12 x + 5) 7. f ( x ) = 7 e 2 x −3 Chapter 3 107 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 8. y  6( x 2  3 x  12)6 9. f ( x)  2 x 2  3x Chapter 3 3 108 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 10. Chapter 3 y  (ln(3 x  7))3 109 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 3 110 Copyright © 2019 Pearson Education, Inc.