Question

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!

Answer 1

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.