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3 Pages

### Derivatives

Course: CALCULUS 135, Spring 2011
School: Rutgers
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Word Count: 902

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ome S Dif f erentiation Rules The following pages list various rules for nding derivatives with very basic examples to show how the rules are used. The following pages are NOT formula sheets for exams or quizzes. The examples are NOT examples or samples of the problems that will be on exams. It is not a substitute for lecture or recitation. Denition: Let f (x) be a function. Then the derivative of f (x) is the...

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ome S Dif f erentiation Rules The following pages list various rules for nding derivatives with very basic examples to show how the rules are used. The following pages are NOT formula sheets for exams or quizzes. The examples are NOT examples or samples of the problems that will be on exams. It is not a substitute for lecture or recitation. Denition: Let f (x) be a function. Then the derivative of f (x) is the function denoted f (x) given by f (x) = lim h0 f (x + h) f (x) h PROVIDED this limit exists. For a particular value of x, say x = a, the derivative evaluated at x = a is given by f (a) = f (x) x=a = lim h0 f (a + h) f (a) h PROVIDED this limit exists at x = a. A function y = f (x) is dierentiable at a if f (a) exists, i.e., if the above limit exists. This value f (a) is called the derivative of f at x = a. df dx Other notations: f (x) d f dx Dx f y dy dx d y dx Dx y Notations for Higher Order Derivatives: 2nd order: f (x ) d2 f dx2 d2 f dx2 (2) Dx f y d2 y dx2 d2 y dx2 (2) Dx y 3rd order: f (x ) d3 f dx3 d3 f dx3 (3) Dx f y d3 y dx3 d3 y dx3 (3) Dx y 4th order: f (4) (x) y (4) f ( n ) (x ) d4 f dx4 dn f dxn (4) Dx f nth order: d4 f dx4 dn f dxn ( Dxn) f y (n) d4 y dx4 dn y dxn d4 y dx4 dn y dxn 1 (4) Dx y ( Dxn) y S ome Dif f erentiation Rules Dif f erentiation Rules b and c are real constants; b > 0, b = 1 Examples ( 5) = 0 1. (c) = 0 (6) = 0 2. (cx) = c x =1 3. (xn ) = nxn1 n = any real number (x8 ) = 8x7 ( 2 ) = 0 (6x) = 6 (x) = 1 x5 = x5 x 5 = = 5x6 1 12 ( 3 x) = (x 3 ) = x 3 3 4. [f (x) g (x)] = f (x) g (x) [x3 + 5x] = (x3 ) + (5x) = 3x2 + 5 5. (4x2 ) = 4(x2 ) = 4(2x) = 8x c f (x ) = cf (x) 6. (ex ) = ex (5x2 + ex ) = 10x + ex 7. (bx ) = bx ln b (3x7 ) = 21x6 (6x) = 6x ln 6 8. (ln x) = 1 x 9. (logb x) = (3 + 2x + ln x) = 2 + 1 x ln b (log5 x) = (3ex ) = 3ex 1 x 1 x ln 5 10. (sin x) = cos x (sin x + ex ) = cos x + ex 11. (cos x) = sin x (5x + cos x) = 5 sin x 12. (tan x) = sec2 x (8x2 + tan x) = 16x + sec2 x 13. (sec x) = sec tan x x (9x + sec x) = 9 + sec x tan x 14. (csc x) = csc x cot x (2x + csc x) = 2x ln 2 csc x cot x 15. (cot x) = csc2 x (4ex + cot x) = 4ex csc2 x 16. Product Rule: [f (x) g (x)] = (x3 sin x) = (3x2 )(sin x) + (x3 )(cos x) f (x ) g (x ) + g (x ) f (x ) f (x ) = g (x ) g (x ) f (x ) f (x ) g (x ) g 2 (x ) 17. Quotient Rule: x3 tan x 2 = (tan x)(3x2 ) (x3 )(sec2 x) tan2 x 1 (x ) 5 = 1 5 Chain Rule: If h(x) = f g (x) = f [g (x)], then h (x) = f g (x ) g (x ) If h(x) = f [g (x)] we can call f (x) the Outer Function and g (x) the inner function. Then h (x) = [Outer (inner )] (inner ) Equivalently, if we write y = h(x) = f (u), where u = g (x), then dy dy du = dx du dx Dif f erentiation Rules w Chain Rule Examples [g (x)n ] = [ng (x)n1 ] g (x) (sin3 x) = [(sin x)3 ] = 3(sin x)2 cos x = 3(sin2 x) cos x 1 2 3 5x3 7x = 5x3 7x 3 = 1 (5x3 7x) 3 (15x2 7) 3 [eg(x) ] = [eg(x) ] g (x) [ecos x ] = [ecos x ] ( sin x) [bg(x) ] = [bg(x) ln b] g (x) [3sin x ] = [3sin x ln 3] (cos x) 1 g (x ) g (x ) 1 [logb g (x)] = g (x ) [g (x)](ln b) [ln(cos x)] = 1 cos x [log5 (tan x)] = [ln g (x)] = 1 (sec2 x) [tan x](ln 5) ( sin x) [sin g (x)] = [cos g (x)] g (x) [sin 4x2 ] = [cos 4x2 ] 8x [cos g (x)] = [ sin g (x)] g (x) [cos(ln x)] = [ sin(ln x)] [tan g (x)] = [sec2 g (x)] g (x) [tan 9x] = [sec2 9x] (9) [sec g (x)] = [sec g (x) tan g (x)] g (x) [sec x4 ] = [sec x4 tan x4 ] (4x3 ) [csc g (x)] = [ csc g (x) cot g (x)] g (x) [csc x7 ] = [ csc x7 cot x7 ] (7x6 ) [cot g (x)] = [ csc2 g (x)] g (x) [cot x5 ] = [ csc 2 x5 ] (5x4 ) 1 x We use the chain rule in a recursive fashion when we have compound compositions such as m(x) = h f g (x) = h f g (x) m (x) = h [f (g (x))] f (g (x)) g (x) Compound Chain Rule Examples: [sin7 (2x3 + 4x)] = [sin(2x3 + 4x)]7 ecos x 2 = ecos x 1 2 = 2 1 cos x e 2 = 7[sin(2x3 + 4x)]6 [cos(2x3 + 4x)] (6x2 + 4) = [7 sin6 (2x3 + 4x)] [cos(2x3 + 4x)] (6x2 + 4) 1 2 (ecos x ) ( sin x) 2 [tan(e4x )] = [sec2 (e4x )] (e4x ) (8x) 3
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Rutgers - CALCULUS - 135
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Rutgers - ANTHROPOLO - 111
Rutgers - ANTHROPOLO - 111
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