Question

find the second derivative of each of the following functions:

a) y=e^x + sin^2 (x)

b) y = 2√(x) + ln (x^3)

Answer 1

To do these, you must have in your repertoire the ability to quickly differentiate
e^x, ln(something), sinx, cosx, and √(something) etc

I will do b) , you try the first one

y = 2√(x) + ln (x^3)
y = 2 x^(1/2) + ln(x^3)
y' = (1/2)(2) x^(-1/2) + 3x^2/x^3
= x^(-1/2) + 3x^-1
y '' = (-1/2)x^(-3/2) - 3x^-2
= -1/(2√x)^3 - 3/x^2 or variations of that after rationalizing the denominator

I got it simplified as
-1/(2x√x) - 3/x^2

Answer 2

for a) i got y' = e^x cos(2x) * 2

for b) i got a different answer, i got y' = (√(x) - 6)) / (2x^2)

i use the symbolab to check my answer and i got it correct?

Answer 3

when simplified, it's the same thing

look at my first term:
-1/(2x√x)
= -1/(2x√x) * √x/√x
= -√x/(2x^2) <---- same as the first term of your answer
and of course my - 3/x^2 is the same as your - 6/(2x^2)

Answer 4

is my answer for a is correct?

Answer 5

a)
y=e^x + sin^2 (x)
y ' = e^x + 2sinx cosx
= e^x + sin (2x)

y'' = e^x + 2cos(2x) , you had them multiplies, could have been just a typo