Question

fins the derivative of:

y=sin(1/x)
y=ln(sqrert(x^2 +1)
y=e^xcos2x
y=x(lnx)^3

Answer 1

Each of these require chain rule

1. sin is the outside function, 1/x is the inside function
2. ln is the outside, sqrt(x^2+1) is the first inside, (x^2+1) is the second inside
3. use product rule. e^x is the first function, cos(2x) is the second function. Don't forget to do chain rule with the 2x
4. again, product rule. x is your first function, (lnx)^3 is the second function. Again, chain rule with the (lnx)^3

If you need help with a specific one, let me know :)

Answer 2

so for 1 the answer would be cos 1/x^3?

Answer 3

and the second one would be x/square root of X^2+1) times x^2+1?

Answer 4

I'll do the first one for you:

d/dx sin(1/x)
= cos(1/x) * d/dx (1/x)
= cos(1/x) * -x^-2
= -cos(1/x)/x^2

Answer 5

For the second:

d/dx ln(sqrt(x^2+1)
=1/sqrt(x^2+1)* d/dx sqrt(x^2+1)
=1/sqrt(x^2+1)* (1/2)(x^2+1)^(-1/2)* d/dx (x^2+1)
=1/sqrt(x^2+1)* (1/2)(x^2+1)^(-1/2)*(2x)
which simplifies to
=x/(x^2+1)

Answer 6

check your answers in wolframalpha if you're not sure