University:
California State University, Northridge
Course:
MATH 351 | Differential Equations
Academic year:

2024
Views:

263

Pages:

3
Author:

Javon Neal

Question 1 What is the antiderivative of 𝑒 𝑥 e x ? Answer Since the derivative of 𝑒 𝑥 e x is 𝑒 𝑥 e x , the antiderivative is: ∫ 𝑒 𝑥 𝑑 𝑥 = 𝑒 𝑥 + 𝐶 ∫e x dx=e x +C Question 2 How do you find the derivative of cot ⁡ 𝑥 cotx? Answer 1 Start with: 𝑦 = cot ⁡ 𝑥 = cos ⁡ 𝑥 sin ⁡ 𝑥 . y=cotx= sinx cosx . Using the quotient rule: 𝑑 𝑑 𝑥 ( 𝑔 ( 𝑥 ) ℎ ( 𝑥 ) ) = 𝑔 ′ ( 𝑥 ) ℎ ( 𝑥 ) − 𝑔 ( 𝑥 ) ℎ ′ ( 𝑥 ) [ ℎ ( 𝑥 ) ] 2 , dx d ( h(x) g(x) )= [h(x)] 2 g ′ (x)h(x)−g(x)h ′ (x) , where 𝑔 ( 𝑥 ) = cos ⁡ 𝑥 g(x)=cosx and ℎ ( 𝑥 ) = sin ⁡ 𝑥 h(x)=sinx. Differentiate: 𝑦 ′ = − sin ⁡ 𝑥 ⋅ sin ⁡ 𝑥 − cos ⁡ 𝑥 ⋅ cos ⁡ 𝑥 ( sin ⁡ 𝑥 ) 2 . y ′ = (sinx) 2 −sinx⋅sinx−cosx⋅cosx . Simplify: 𝑦 ′ = − sin ⁡ 2 𝑥 − cos ⁡ 2 𝑥 sin ⁡ 2 𝑥 = − 1 sin ⁡ 2 𝑥 = − csc ⁡ 2 𝑥 . y ′ = sin 2 x −sin 2 x−cos 2 x = sin 2 x −1 =−csc 2 x. Answer 2 Alternatively, write cot ⁡ 𝑥 = cos ⁡ 𝑥 / sin ⁡ 𝑥 cotx=cosx/sinx and use the quotient rule directly. The result remains: 𝑑 𝑑 𝑥 ( cot ⁡ 𝑥 ) = − csc ⁡ 2 𝑥 . dx d (cotx)=−csc 2 x. Question 3 Find the antiderivative of csc ⁡ 2 𝑥 csc 2 x. Answer 1 We know that the derivative of cot ⁡ 𝑥 cotx is: 𝑑 𝑑 𝑥 ( cot ⁡ 𝑥 ) = − csc ⁡ 2 𝑥 . dx d (cotx)=−csc 2 x. Thus, the antiderivative of csc ⁡ 2 𝑥 csc 2 x is: ∫ csc ⁡ 2 𝑥 𝑑 𝑥 = − cot ⁡ 𝑥 + 𝐶 . ∫csc 2 xdx=−cotx+C. Answer 2 Using the common integral formula: ∫ csc ⁡ 2 𝑥 𝑑 𝑥 = − cot ⁡ 𝑥 + 𝐶 . ∫csc 2 xdx=−cotx+C. Question 4 What is the derivative of sin ⁡ − 1 ( 𝑥 ) sin −1 (x)? Answer 1 Let 𝑦 = sin ⁡ − 1 ( 𝑥 ) y=sin −1 (x). Then sin ⁡ 𝑦 = 𝑥 siny=x and − 𝜋 2 ≤ 𝑦 ≤ 𝜋 2 − 2 π ≤y≤ 2 π . Differentiating implicitly: cos ⁡ ( 𝑦 ) 𝑑 𝑦 𝑑 𝑥 = 1. cos(y) dx dy =1. Since cos ⁡ 2 ( 𝑦 ) = 1 − sin ⁡ 2 ( 𝑦 ) cos 2 (y)=1−sin 2 (y), we get: cos ⁡ ( 𝑦 ) = 1 − 𝑥 2 . cos(y)= 1−x 2 . Thus: 𝑑 𝑦 𝑑 𝑥 = 1 1 − 𝑥 2 . dx dy = 1−x 2 1 . Answer 2 Using the standard derivative formula for the inverse sine function: 𝑑 𝑑 𝑥 ( sin ⁡ − 1 ( 𝑥 ) ) = 1 1 − 𝑥 2 . dx d (sin −1 (x))= 1−x 2 1 . Question 5 What is the derivative of log ⁡ ( 𝑥 ) log(x)? Answer 1 The derivative depends on the logarithmic base. For natural logarithms: 𝑑 𝑑 𝑥 ( log ⁡ 𝑒 𝑥 ) = 1 𝑥 . dx d (log e x)= x 1 . For logarithms with base 𝑏 b: 𝑑 𝑑 𝑥 ( log ⁡ 𝑏 𝑥 ) = 1 𝑥 ln ⁡ 𝑏 . dx d (log b x)= xlnb 1 . Answer 2 For a logarithm with general base 𝑏 b, where 𝑓 ( 𝑥 ) = log ⁡ 𝑏 ( 𝑥 ) f(x)=log b (x): 𝑓 ′ ( 𝑥 ) = 1 𝑥 ln ⁡ ( 𝑏 ) . f ′ (x)= xln(b) 1 . For example, for log ⁡ 2 ( 𝑥 ) log 2 (x): 𝑓 ′ ( 𝑥 ) = 1 𝑥 ln ⁡ ( 2 ) . f ′ (x)= xln(2) 1 .

Derivatives #24 - Questions and Answers

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