Question

find the first derivative of y = 4cos(sin3x)

Answer 1

y=4cos(sin3x)

Find the derivative of the expression.
(d)/(dx) 4cos(sin3x)

The chain rule states that the derivative of a composite function (f o g)' is equal to (f' o g)*g'. To find the derivative of 4cos(sin3x), find the derivatives of each portion of the function and use the chain rule formula.
(d)/(du) 4cos(u)*(d)/(dx) sin3x

The derivative of 4cos(u) is -(4sin(u)).
(d)/(du) 4cos(u)=-(4sin(u))

Multiply -1 by the 4sin(u) inside the parentheses.
(d)/(du) 4cos(u)=-4sin(u)

The chain rule states that the derivative of a composite function (h o j)' is equal to (h' o j)*j'. To find the derivative of sin3x, find the derivatives of each portion of the function and use the chain rule formula.
(d)/(du) sin(u)*(d)/(dx) 3x

The derivative of sin(u) is cos(u).
(d)/(du) sin(u)=cos(u)

To find the derivative of 3x, multiply the base (x) by the exponent (1), then subtract 1 from the exponent (1-1=0). Since the exponent is now 0, x is eliminated from the term.
(d)/(dx) 3x=3

Replace the variable u with 3x in the expression.
(d)/(du) sin(u)=cos((3x))

Replace the variable u with 3x in the expression.
3

Form the derivative by substituting the values for each portion into the chain rule formula.
=cos((3x))*3

Multiply cos((3x)) by 3 to get 3cos((3x)).
(d)/(dx) sin3x=3cos((3x))

Replace the variable u with sin3x in the expression.
(d)/(du) 4cos(u)=-4sin((sin3x))

Replace the variable u with sin3x in the expression.
3cos((3x))

Form the derivative by substituting the values for each portion into the chain rule formula.
=-4sin((sin3x))*3cos((3x))

Multiply -4sin((sin3x)) by 3cos((3x)) to get -12sin((sin3x))cos((3x)).
(d)/(dx) 4cos(sin3x)=-12sin((sin3x))cos((3x))

Answer 2

that's a very long reply O.o. thank you for your efforts!
my answer is -12cos(sin3x)sin(sin3x)cos(3x)

Answer 3

but im not really sure

Answer 4

\[y = 4Cos(Sin(3x))\]\[y'=4(-Sin(Sin(3x)))*(Sin(3x))'<-\space Chain \space Rule\]

Answer 5

now just differentiate the second bit:
\[(Sin(3x))' = Cos(3x)*(3x)'=Cos(3x)*3=3Cos(3x)\]

Answer 6

Again I used the chain rule.

Answer 7

So now putting the two equations together, we get:
\[y'=4(-Sin(Sin(3x)))*(3Cos(3x))=-12Sin(Sin(3x))*Cos(3x)\]

Answer 8

thank you! I saw my mistake
my answer is -12sin(sin3x)cos(3x)
and then i saw your reponse
thank you for valiant effort!

Answer 9

Yep looks good! Nice work @bahrom7893

Answer 10

Cheers