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Common Derivatives and Integrals You can navigate to specific sections of this handout by clicking the links below. Derivative Rules: pg. 1 Integral Formulas: pg. 3 Derivatives Rules for Trigonometric Functions: pg. 4 Integrals of Trigonometric Functions: pg. 5 Special Differentiation Rules: pg. 6 Special Integration Formulas: pg. 7 Derivative Rules: d [ ] ′ 1. Constant Multiple Rule dx cu = cu , where c is a constant. d [ ] ′ ′ 2. Sum and Difference Rule dx u ± v = u ± v d [ ] ′ ′ 3. Product Rule dx uv = uv + vu 4. Quotient Rule d u = vu′−uv′ dx v v2 d [ ] 5. Constant Rule, dx c = 0 d [ n]= n−1 ′ 6. Power Rule dx u nu u d [ ] 7. Power Rule dx x =1 8. Derivative Involving Absolute Value d [ ] u ( ′) dx u = u u ,u ≠ 0 d [ ] u′ 9. Derivative of the Natural Logarithmic Function dx lnu = u Common Derivatives and Integrals Provided by the Academic Center for Excellence 1 Reviewed June 2008 d [ u]= u ′ 10. Derivative of Natural Exponential Function dx e e u ( ) ( 2 )( 2 ) Example 1: Find the derivative of f x = 4x −3x 3+2x Since there are two polynomials multiplied by each other, apply the third derivative rule, the Product Rule, to the problem. This is the result of the Product Rule: ′ 2 d [ 2 ] 2 d [ 2 ] ( ) ( ) ( ) f x = 4x−3x dx 3+2x + 3+2x dx 4x−3x Now, take the derivative of each term inside of the brackets. Multiple derivative rules are used, including the Sum and Difference Rule, Constant Rule, Constant Multiple Rule, and Power Rule. When applied, the result is: ′ 2 2 ( ) ( ) ( ) ( ) ( ) f x = 4x−3x 0+4x + 3+2x 4−6x Simplify: ′ 2 2 ( ) ( ) ( ) ( ) ( ) f x = 4x−3x 4x + 3+2x 4−6x Multiply the polynomials by each other: ′ 2 3 2 3 ( ) ( ) ( ) f x = 16x −12x + 12+8x −18x−12x ′ 2 3 2 3 f (x)=16x −12x +12+8x −18x−12x Combine like terms to get a simplified answer: ′ 3 2 f (x)= −24x +24x −18x+12 Common Derivatives and Integrals Provided by the Academic Center for Excellence 2 Reviewed June 2008 Integral Formulas: Indefinite integrals have +C as an arbitrary constant. ( ) ( ) 1.∫kf u du = k∫ f u du, where k is a constant. ( ) ( ) ( ) ( ) [ ] 2. ∫ f u ± g u du = ∫ f u du ± ∫ g u du 3. ∫ du = u + C n+ u 1 4. ∫undu = +C,n≠−1 n+1 5. ∫ du = lnu +C u 6. ∫eudu = eu +C ( 2 3 ) Example 2: Evaluate ∫ 4x −5x +12 dx To evaluate this problem, use the first four Integral Formulas. First, use formula 2 to make the large integral into three smaller integrals: ( 2 3 ) 2 3 ∫ 4x −5x +12dx=∫4x dx−∫5x dx+∫12dx Second, pull out the constants by using formula 1: =4∫x2dx−5∫x3dx+12∫dx Now find each integral using formulas 3 and 4: x2+1 x3+1 ( ) =4 −5 +12 x +C 2 +1 3+1 Provided by the Academic Center for Excellence 3 Common Derivatives and Integrals Although three integrals have been removed, only one constant C is needed because C represents all unknown constants. Therefore multiple C’s can be combined into just one C. To get the final answer, simplify the expression: x3 x4 =4 −5 +12x+C 3 4 =4x3 −5x4 +12x+C 3 4 Derivatives Rules for Trigonometric Functions: d ( ) ( ( )) ′ [ ] 1. dx sin u = cos u u d ( ) ( ( )) ′ [ ] 2. dx cos u =−sin u u d ( ) 2( ) ′ [ ] ( ) 3. dx tan u = sec u u d ( ) 2( ) ′ [ ] ( ) 4. dx cot u =−csc u u d ( ) ( ( ) ( )) ′ [ ] 5. dx sec u = sec u tan u u d ( ) ( ( ) ( )) ′ [ ] 6. dx csc u =−cscu cot u u ( ) Example 3: Find the derivative of f (x)= sin x cos(x) When finding the derivatives of trigonometric functions, non-trigonometric derivative rules are often incorporated, as well as trigonometric derivative rules. Looking at this function, one can see that the function is a quotient. Therefore, use derivative rule 4 on page 1, the Quotient Rule, to start this problem: Provided by the Academic Center for Excellence 4 Common Derivatives and Integrals
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