To simplify step 1 into step 2, your instructor used the quotient rule of differentiation. Let's break down the steps:
Step 1:
- Start with the expression (y''x-y'')/((1+(y')^2)^(3/2))
- Use the product rule of differentiation to expand y''x. This gives: y''x + y'x'
- Substitute y'x' instead of y''x in the numerator, giving: y'x' - y''
- Keep the denominator the same: (1+(y')^2)^(3/2)
- Factor out y': y'(x'/(sqrt(1+(y')^2)) - y)
- Simplify and rearrange terms: (y'((x'(1+(y')^2)^(3/2)) - y(1+(y')^2)))/(1+(y')^2)^(3/2)
Step 2:
- Start with the expression (y'x' - y)/(1+(y')^2)^(3/2)
- Perform the same factorization of y' as in step 1: y'(x'/(sqrt(1+(y')^2)) - y)
- Simplify and rearrange terms: y'*(y'' + ((y')^2)/(sqrt(1+(y')^2))))/(1+(y')^2)^(3/2) - y*((y')^2)/(1+(y')^2)^(3/2) - sqrt(1+(y')^2)
As you can see, steps 1 and 2 are equivalent. The only difference is in how the expressions are presented.
My instructor has given the question in the image and the sample answer has also been given by him.
Image of the question and the instructor's answer: drive(DOT)google(DOT)com/file/d/1XkjhzrN1MKNc0XDrLlAR_pu6l39P9AY7/view?usp=share_link
However, I don't understand how step 1 simplifies into step 2, as in his calculation. Instead, I get below as step 2.
Step 2 according to my calculations :
( y*(y'') + ((y')^2) )/(sqrt(1+((y')^2)))) - (y*((y')^2)/(1+((y')^2))^(3/2)) - (sqrt((1+((y')^2))))
Any help in understanding this will be highly appreciated!
1 answer
1 answer