Answer:
To differentiate the given expression, we will need to use both the product rule and the quotient rule.
Let's first rewrite the expression using the reciprocal identity for cosine:
sin y/cos x = sin y * (cos x)^(-1)
Now we can use the product rule and the quotient rule to find the derivative with respect to m:
d/dm [sin y * (cos x)^(-1)] = sin y * d/dm[(cos x)^(-1)] + (cos x)^(-1) * d/dm[sin y]
Using the chain rule, we can find the derivative of (cos x)^(-1) with respect to m:
d/dm[(cos x)^(-1)] = -1/(cos^2 x) * (-sin x) * (dx/dm)
d/dm[(cos x)^(-1)] = sin x/cos^2 x * (dx/dm)
Using the chain rule again, we can find the derivative of sin y with respect to m:
d/dm[sin y] = cos y * (dy/dm)
Substituting these expressions back into the original equation, we get:
d/dm [sin y/cos x] = sin y * [sin x/cos^2 x * (dx/dm)] + cos y * (dy/dm) * (cos x)^(-1)
Simplifying this expression, we get:
d/dm [sin y/cos x] = (sin y * sin x/cos^2 x) * (dx/dm) + (cos y/cos x) * (dy/dm)
Therefore, the derivative of sin y/cos x with respect to m is (sin y * sin x/cos^2 x) * (dx/dm) + (cos y/cos x) * (dy/dm).