If Y=sin2x/cosx than dy/dx is
using the quotient rule,
dy/dx = [2cos2x * cosx - sin2x(-sinx)]/cos^2x
= [2(1-2sin^2x)(cosx) + (2sinx cosx)(sinx)]/cos^2x
= [2cosx - 4sin^2x cosx + 2sin^2x cosx]/cos^2x
= (2cosx - 2sin^2x cosx)/cos^2x
= 2cosx(1-sin^2x)/cos^2x
= 2cosx cos^2x/cos^2x
= 2cosx
Of course, you could avoid all that mess by noting that
sin2x/cosx = (2sinx cosx)/cosx = 2sinx
To find the derivative of Y with respect to x (dy/dx), we can use the quotient rule, as the expression Y is a ratio of two functions: sin(2x) and cos(x).
The quotient rule states that for a function u(x) divided by another function v(x), the derivative du/dx is given by:
dy/dx = (v(x) * du/dx - u(x) * dv/dx) / [v(x)]^2
Let's apply the quotient rule to the given expression Y = sin(2x) / cos(x):
First, we identify u(x) and v(x):
u(x) = sin(2x)
v(x) = cos(x)
Next, we find the derivatives du/dx and dv/dx:
du/dx = derivative of sin(2x) = 2*cos(2x)
dv/dx = derivative of cos(x) = -sin(x)
Now, we plug these values into the quotient rule formula:
dy/dx = (cos(x) * 2*cos(2x) - sin(x) * sin(2x)) / [cos(x)]^2
Finally, we simplify the expression further if necessary.