You use the chain rule to solve this problem.
First you can separate the terms that are added, because the
derivative of a sum is the sum of the derivatives, so
dy/dx = d/dx(sin3x) + d/dx(cos7x)
Now you work with these derivatives separately. Let's look at the first one:
sin3x is a function of x within a function (3x is the inner function and sin() is the outer function). The chain rule says that to take the derivative of this with respect to the inside variable x, you take the derivative of the outside function with respect to the inside times the derivative of the inside with respect to the variable x.
So let A = 3x. Then
d/dx(sin3x) = d/dx(sinA) = ( d/dA(sinA)) ( d/dx(A))
= cosA ( d/dx(3x) = cosA (3) = 3cosA.
Now do the same steps for the other addend and you'll have your answer.
Stephen La Rocque.