Find the Derivative - d/dx cos(x-y)

Problem

d(cos(x−y))/d(x)

Solution

1. Identify the need for implicit differentiation and the chain rule, as y is assumed to be a function of x

2. Apply the chain rule to the outer function, which is the cosine function.

d(cos(u))/d(u)=−sin(u)

3. Differentiate the inner function x−y with respect to x

d(x−y)/d(x)=1−d(y)/d(x)

4. Combine the results using the chain rule formula.

−sin(x−y)⋅(1−d(y)/d(x))

5. Distribute the sine term to simplify the expression.

−sin(x−y)+sin(x−y)d(y)/d(x)

Final Answer

d(cos(x−y))/d(x)=sin(x−y)d(y)/d(x)−sin(x−y)

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