Find the Derivative - d/dx cos(xy)

Problem

d(cos(x*y))/d(x)

Solution

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

d(cos(u))/d(u)=−sin(u)

2. Identify the inner function as u=x*y and prepare to multiply by its derivative with respect to x

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

3. Apply the product rule to the inner function x*y treating y as a function of x

(d(x)*y)/d(x)=x⋅d(y)/d(x)+y⋅d(x)/d(x)

4. Simplify the derivative of the inner function.

(d(x)*y)/d(x)=xd(y)/d(x)+y

5. Combine the results by substituting the derivative of the inner function back into the chain rule expression.

d(cos(x*y))/d(x)=−sin(x*y)*(xd(y)/d(x)+y)

6. Distribute the term to reach the final form.

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

Final Answer

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

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