Differentiation Shortcuts Sections 10-5, 11-2, 11-3, and 11-4 Prof. Nathan Wodarz Math 109 - Fall 2008 Contents 1 Basic Properties 2 1.1 Notation............................... 2 1.2 Constant Functions......................... 2 1.3 Power Rule............................. 3 1.4 Sum and Difference Properties................... 3 1.5 Constant Multiple Property..................... 4 1.6 Basic Property Practice....................... 5 2 Exponential and Logarithmic Functions 6 2.1 Derivative of e x........................... 6 2.2 Derivative of ln x.......................... 7 2.3 Other Exponential and Logarithmic Functions........... 7 3 Derivatives of Products and Quotients 8 3.1 The Product Rule.......................... 8 3.2 The Quotient Rule.......................... 9 4 The Chain Rule 10 4.1 Composite Functions........................ 10 4.2 General Power Rule......................... 12 4.3 Chain Rule For Exponential Functions............... 12 4.4 Chain Rule For Logarithmic Functions............... 13 4.5 General Chain Rule......................... 13 1
1 Basic Properties 1.1 Notation Notation The following are all notations to represent the derivative of y = f (x) f (x) y dy dx 1.2 Constant Functions Constant Functions If y = f (x) = C, then f (x) = 0 Also write as y = 0 dy/dx = 0 C = 0 d dx C = 0 Constant Functions Problem 1. Find y if y = 5 8 A. 0 B. 5 8 C. 1 D. 5 8 x 2
1.3 Power Rule Power Rule If y = f (x) = x n (n a real number), then f (x) = nx n 1 Also write as y = nx n 1 dy/dx = nx n 1 Power Rule Problem 2. Find y if y = x 6 A. x 5 B. 6x 4 C. 6x 5 D. 6x 7 1.4 Sum and Difference Properties Sum and Difference Properties If y = f (x) = u(x) ± v(x), then f (x) = u (x) ± v (x) Also write as y = u ± v dy dx = du dx + dv dx Use the same sign in the derivative as in the original function 3
Sum and Difference Properties Problem 3. Find f (x) if f (x) = x 2 + x 3 + x A. 2x 3 + 3x 2 B. 2x 3 + 3x 2 + 1 C. 2x 1 + 3x 2 + 1 D. 2x 1 + 3x 2 1.5 Constant Multiple Property Constant Multiple Property If y = f (x) = ku(x), then f (x) = ku (x) Also write as y = ku dy = k du dx dx Constant Multiple Property Problem 4. Let f and g be functions with f (4) = 2 and g (4) = 3. Find h (4) for h(x) = 3 f (x) g(x) + 2 A. 2 B. 5 C. 9 D. 11 4
1.6 Basic Property Practice Basic Property Practice Problem 5. Find d dv (6v0.7 v 5.8 ) A. 4.2v 0.3 5.8v 4.7 B. 4.2v 0.3 5.8v 4.7 C. 4.2v 0.3 5.8v 4.8 D. 4.2v 0.3 5.8v 4.8 Basic Property Practice Problem 6. Find f (x) if f (x) = 3π 6 A. 0 B. 3 C. 3π 5 D. 18π 5 Basic Property Practice Problem 7. Find y if y = 4 x 4 5 3 x A. 1 x 3 + 5 3 x 4/3 B. 1 4x 3 5 3 x 2/3 C. 1 4 x 5 15x 2/3 D. 16x 5 5 3 x 2/3 5
Basic Property Practice Problem 8. Find an equation of the tangent line at x = 2 for f (x) = 4+x 2x 2 3x 3 A. y = 47x + 68 B. y = 43x + 48 C. y = 43x + 60 D. y = 39x + 52 2 Exponential and Logarithmic Functions 2.1 Derivative of e x Derivative of e x If y = f (x) = e x, then f (x) = e x WARNING! The derivative is not xe x 1 Derivative of e x Problem 9. Find the derivative of f (x) = 5e x 4x 8 and simplify. A. f (x) = 5e x 32x 7 B. f (x) = 5e x 32x 8 C. f (x) = 5e x 4x 8 D. f (x) = 5e 5x 32x 7 6
2.2 Derivative of ln x Derivative of ln x If y = f (x) = ln x, then f (x) = 1 x Can use earlier properties to take more complicated derivatives, such as Derivative of ln x d dx ln xn = d dx n ln x = n x Problem 10. Find y for y = ln 6x 2 A. y = 2 x B. y = 12 x C. y = 2x x 2 +6 D. y = 1 2x+6 2.3 Other Exponential and Logarithmic Functions Other Exponential Functions If b > 0, b 1: d dx bx = b x ln b Other Logarithmic Functions If b > 0, b 1: d dx log b x = 1 ln b ( 1 x) 7
Other Exponential and Logarithmic Functions Problem 11. Find f (x) for f (x) = 8 x + 4 log 3 (x 2 ) A. f (x) = 8 x + 8 x B. f (x) = 8 x + 8 x ln 3 C. f (x) = 8 x ln 8 + 8 x ln 3 D. f (x) = 8(7 x ) + 4 x 2 3 Derivatives of Products and Quotients 3.1 The Product Rule The Product Rule If y = f (x) = F(x)S (x) where F and S are differentiable, then f (x) = F(x)S (x) + S (x)f (x) Also written as: y = FS + S F The Product Rule Problem 12. Find f (x) for f (x) = (5x 3 + 4)(3x 7 5) A. f (x) = 20x 9 + 84x 6 75x B. f (x) = 20x 9 + 84x 6 75x 2 C. f (x) = 150x 9 + 84x 6 75x D. f (x) = 150x 9 + 84x 6 75x 2 8
Constant Multiple Property Problem 13. Let f and g be functions that satisfy f (2) = 1, g(2) = 3, f (2) = 2, and g (2) = 3. Find h (2) for h(x) = f (x)g(x) 2 f (x) + 7 A. 6 B. 5 C. 5 D. 6 3.2 The Quotient Rule The Quotient Rule If y = f (x) = T(x) where T and B are differentiable, then B(x) Also written as: y = BT T B B 2 The Quotient Rule Problem 14. Find f (t) for f (t) = A. 5 7t 5 B. 5 (7t 5) 2 C. 5t (7t 5) 2 D. 14t 5 (7t 5) 2 f (x) = B(x)T (x) T(x)B (x) [B(x)] 2 t 7t 5 9
The Quotient Rule Problem 15. Find f (2) if f (x) = 2x 7 3x 2 A. 17 4 B. 17 16 C. 17 4 D. 17 16 The Quotient Rule Problem 16. Acme Corporation Publishing House has started publishing a new magazine for college students. The monthly sales S (in thousands) are given by S (t) = 800t where t is the number of months since the first issue was published. t+2 Find S (3) and S (3) and interpret the results. A. At three months, monthly sales are 480,000 and decreasing at 64,000 magazines per month B. At three months, monthly sales are 480,000 and increasing at 64,000 magazines per month C. At three months, monthly sales are 2,400,000 and increasing at 64,000 magazines per month D. At three months, monthly sales are 2,400,000 and increasing at 800,000 magazines per month 4 The Chain Rule 4.1 Composite Functions Composite Functions A function like f (x) = 1 + x 2 can be built by chaining two simpler functions together 10
Take y = f (u) = u and u = g(x) = 1 + x 2 The composite of f and g expresses y as a function of x: y = f (u) = f (1 + x 2 ) The composite of functions f and g is the function m(x) = f [g(x)] Composite Functions Problem 17. Find the composition f [g(x)] if f (u) = u 5 and g(x) = 2 3x 2 A. (2 3x 2 ) 2 B. (2 3x 2 ) 5 C. (10 15x 2 ) 5 D. 2 3u 10 Composite Functions Problem 18. Choose the answer choice that includes all pairs of functions from 1 the list so that the composite function h(x) = can be written in the form 11 x 2 h(x) = f [g(x)] A. f (x) = 1 x and g(x) = 11 x 2 B. f (x) = 11 x and g(x) = 1 x 2 C. f (x) = 1 11 x and g(x) = x 2 D. Both A and C 11
4.2 General Power Rule General Power Rule If u(x) is a differentiable function,n is any real number, and y = f (x) = [u(x)] n, then f (x) = n[u(x)] n 1 u (x) General Power Rule Problem 19. Find d 4 dω (ω 2 + 3) 5 A. 40 (ω 2 +3) 6 B. 40x (ω 2 +3) 6 C. 40 (ω 2 +3) 5 D. 40 (ω 2 +3) 6 General Power Rule Problem 20. Find dy dx if y = 8 8x7 10 A. 8 7 8x7 10 B. 448x 6 7 8x7 10 C. 7x 6 (8x 7 10) 7/8 D. 56x 6 (8x 7 10) 7/8 4.3 Chain Rule For Exponential Functions Chain Rule For Exponential Functions If u(x) is a differentiable function and y = f (x) = e u(x), then f (x) = e u(x) u (x) 12
General Power Rule Problem 21. Find f (x) if f (x) = 3 x 1 A. 3 ln(3) B. 3 x 1 ln x C. 3 x 1 ln 3 D. 3 x 1 ln(3 x 1 ) 4.4 Chain Rule For Logarithmic Functions Chain Rule For Logarithmic Functions If u(x) is a differentiable function and y = f (x) = ln[u(x)], then f (x) = u (x) u(x) General Power Rule Problem 22. Find y if y = log 7 (x 6 + 1) A. 6x 5 (ln 7)(x 6 +1) B. 6x 5 (ln 7) x 6 +1 C. 6x 5 x 6 +1 D. 1 (ln 7)(x 6 +1) + 6x5 4.5 General Chain Rule General Chain Rule If y = f (u) and u = g(x) give a composite function y = m(x) = f [g(x)], then dy = dy du, if both derivatives on the right exist dx du dx m (x) = f [g(x)]g (x), if both derivatives on the right exist 13
General Chain Rule Problem 23. Find all values of x where the tangent line to f (x) = horizontal. x (x 2 + 3) 3 is A. x = ± 15 5 B. x = 0, ± 15 5 C. x = ± 3 5 D. x = 0 Summary Summary You should be able to: Use basic properties to compute simple derivatives Find the derivatives of exponential and logarithmic functions Use the product and quotient rules Use the chain rule to differentiate composite functions 14