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

2024
Views:

444

Pages:

3
Author:

Javon Neal

Question 1 What is the derivative of 𝑥 𝑒 𝑥 xe x ? Answer 1 To evaluate the derivative of 𝑥 𝑒 𝑥 xe x , use the product rule: 𝑑 𝑑 𝑥 [ 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) ] = 𝑓 ′ ( 𝑥 ) 𝑔 ( 𝑥 ) + 𝑓 ( 𝑥 ) 𝑔 ′ ( 𝑥 ) dx d [f(x)g(x)]=f ′ (x)g(x)+f(x)g ′ (x) Here: 𝑓 ( 𝑥 ) = 𝑥 , 𝑓 ′ ( 𝑥 ) = 1 , 𝑔 ( 𝑥 ) = 𝑒 𝑥 , 𝑔 ′ ( 𝑥 ) = 𝑒 𝑥 f(x)=x,f ′ (x)=1,g(x)=e x ,g ′ (x)=e x Substitute into the product rule: 𝑑 𝑑 𝑥 [ 𝑥 𝑒 𝑥 ] = ( 1 ) ( 𝑒 𝑥 ) + ( 𝑥 ) ( 𝑒 𝑥 ) dx d [xe x ]=(1)(e x )+(x)(e x ) Simplify: 𝑑 𝑑 𝑥 [ 𝑥 𝑒 𝑥 ] = 𝑒 𝑥 + 𝑥 𝑒 𝑥 = 𝑒 𝑥 ( 𝑥 + 1 ) dx d [xe x ]=e x +xe x =e x (x+1) Answer 2 Using the product rule directly: 𝑑 𝑑 𝑥 [ 𝑥 𝑒 𝑥 ] = 𝑢 ′ 𝑣 + 𝑢 𝑣 ′ , dx d [xe x ]=u ′ v+uv ′ , where 𝑢 = 𝑥 u=x and 𝑣 = 𝑒 𝑥 v=e x . Then, 𝑢 ′ = 1 u ′ =1 and 𝑣 ′ = 𝑒 𝑥 v ′ =e x : 𝑑 𝑑 𝑥 [ 𝑥 𝑒 𝑥 ] = 1 ⋅ 𝑒 𝑥 + 𝑥 ⋅ 𝑒 𝑥 = 𝑒 𝑥 + 𝑥 𝑒 𝑥 = 𝑒 𝑥 ( 𝑥 + 1 ) dx d [xe x ]=1⋅e x +x⋅e x =e x +xe x =e x (x+1) Question 2 What is the derivative of 5 𝑥 5 x ? Answer 1 The general rule for the derivative of 𝑎 𝑥 a x is: 𝑑 𝑑 𝑥 [ 𝑎 𝑥 ] = 𝑎 𝑥 ln ⁡ 𝑎 where 𝑎 > 0 . dx d [a x ]=a x lnawhere a>0. For 𝑎 = 5 a=5: 𝑑 𝑑 𝑥 [ 5 𝑥 ] = 5 𝑥 ln ⁡ 5. dx d [5 x ]=5 x ln5. Answer 2 To confirm: Let 𝑦 = 5 𝑥 y=5 x . Take the natural logarithm: ln ⁡ 𝑦 = 𝑥 ln ⁡ 5. lny=xln5. Differentiate both sides implicitly with respect to 𝑥 x: 1 𝑦 𝑑 𝑦 𝑑 𝑥 = ln ⁡ 5. y 1 dx dy =ln5. Multiply by 𝑦 = 5 𝑥 y=5 x : 𝑑 𝑦 𝑑 𝑥 = 5 𝑥 ln ⁡ 5. dx dy =5 x ln5. Question 5 Find the derivative of the following function 𝑓 ( 𝑥 ) = 10 𝑥 4 − 32 𝑥 + 1 2 f(x)=10x 4 −32x+ 2 1 . Answer To differentiate 𝑓 ( 𝑥 ) = 10 𝑥 4 − 32 𝑥 + 1 2 f(x)=10x 4 −32x+ 2 1 : Use the power rule: 𝑑 𝑑 𝑥 [ 𝑥 𝑛 ] = 𝑛 𝑥 𝑛 − 1 . dx d [x n ]=nx n−1 . Term-by-term differentiation: For 10 𝑥 4 10x 4 : 𝑑 𝑑 𝑥 [ 10 𝑥 4 ] = 10 ⋅ 4 𝑥 3 = 40 𝑥 3 . dx d [10x 4 ]=10⋅4x 3 =40x 3 . For − 32 𝑥 −32x: 𝑑 𝑑 𝑥 [ − 32 𝑥 ] = − 32. dx d [−32x]=−32. For 1 2 2 1 : 𝑑 𝑑 𝑥 [ 1 2 ] = 0. dx d [ 2 1 ]=0. Combine the results: 𝑓 ′ ( 𝑥 ) = 40 𝑥 3 − 32. f ′ (x)=40x 3 −32. Final Answer: 𝑓 ′ ( 𝑥 ) = 40 𝑥 3 − 32. f ′ (x)=40x 3 −32.