Find the Derivative - d/dx sin(xy)

Problem

d(sin(x*y))/d(x)

Solution

1. Identify the rule needed for the expression, which is the chain rule because the function sin(x*y) has an inner function x*y

2. Apply the chain rule by taking the derivative of the outer function sin(u) with respect to u which is cos(u) and then multiplying by the derivative of the inner function x*y with respect to x

3. Apply the product rule to the inner function x*y to find its derivative with respect to x

4. Differentiate the product x*y using the formula (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) where u=x and v=y

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

5. Simplify the derivative of the inner function.

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

6. Combine the results from the chain rule and the product rule.

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

Final Answer

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

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