Sinx=e^y , 0 <x <pi, what is dy/dx in terms of x?
cos x dx = e^y dy
dy/dx = cos x /e^y
but e^y = sin x
so dy/dx = cos x / sin x = cot x
cotx
To find dy/dx in terms of x, we'll need to differentiate both sides of the equation with respect to x. Let's start by differentiating sin(x) = e^y.
To find d/dx(sin(x)), we can use the derivative of the sine function, which is cos(x).
To find d/dx(e^y), we can use the chain rule. The derivative of e^y with respect to y is simply e^y, and since we are differentiating with respect to x, we multiply it by dy/dx.
Applying the chain rule, we get:
cos(x) = e^y * dy/dx
Now, we can solve for dy/dx by rearranging the equation:
dy/dx = cos(x) / e^y
Since we initially had sin(x) = e^y, we can substitute e^y with sin(x) in the equation:
dy/dx = cos(x) / sin(x)
So, the derivative dy/dx in terms of x is cos(x) / sin(x).