Find dy/dx for y=sin(x+y)
A. 0
B. (cos(x+y))/(1-cos(x+y)
C. cos(x+y)
D. 1
E. None of these
I know I'm supposed to use implicit differentiation but I'm not sure how to go about it with sin
y = sin(x+y)
y' = cos(x+y)(1+y')
y' = cos(x+y) + y'*cos(x+y)
y' (1-cos(x+y)) = cos(x+y)
so, (B)
To find dy/dx using implicit differentiation, we treat y as a function of x and differentiate both sides of the equation with respect to x. In this case, we have y = sin(x + y).
Let's start by differentiating both sides of the equation:
d/dx(y) = d/dx(sin(x + y))
The derivative of y with respect to x is simply dy/dx, so we can rewrite the left side as dy/dx:
dy/dx = d/dx(sin(x + y))
To simplify the right side, we use the chain rule. The chain rule tells us that the derivative of a composition of functions is the derivative of the outer function multiplied by the derivative of the inner function. In this case, the outer function is sin(x + y), and the inner function is x + y.
The derivative of sin(x + y) with respect to its argument (x + y) is cos(x + y). Therefore, we can rewrite the right side as:
dy/dx = cos(x + y)
So, the correct answer is C. cos(x + y).